From the NCTM…
   Middle school should acquire a deep 
    understanding of fractions and be able to 
    use them competently in problem solving. 
   NCTM(2000)
   From the NAEP…
   Reports show that fractions are 
     "exceedingly difficult for children to 
     master. "
   Students are frequently unable to 
     remember prior experiences about 
     fractions covered in lower grade levels
   NAEP, 2001
          National Assessment of Educational Progress 
   Mathematics Proficiency
   Conceptual understanding 
   Procedural fluency
   Strategic competence
   Adaptive reasoning
   Productive disposition
    Adding it Up, - National Research Council 
   Bottlenecks in K-8
   It is widely recognized that there are at 
    least two major bottlenecks in the 
    mathematics education of grades K–8: 
     The teaching of fractions
     The introduction of algebra
    Student mistakes with fractions
    Algorithmically based mistakes
    Intuitively based mistakes
    Mistakes based on formal knowledge.
    e.g. Children may try to apply ideas they 
      have about whole numbers to rational 
        Tirosh (2000) 
      numbers and run into trouble