The Trouble with Fractions

The Four Big Ideas of Fractions The parts are of equal size There are a specific number of parts The whole is divided The parts equal the whole

Fractions: Equivalence Fractions with different numbers can be equal one whole 1/8 1/2 1/4 = 4/4 = 2/2 = 8/8 1/2 1/4 1/8 1/1

Fractions: Non-Equivalence If the whole = 8 1/2 of 2 = 1 1/2 of 8 = 4 If the whole = 2 Does 1/2 equal 1/2? One-half of what?

Fractions: Symmetry of Area Recognizing whether shapes have same size (i.e. equal parts)

Fractions: Confusing Names The larger the number (denominator), the smaller the quantity. 1/12 1/2

Fractions: Visual Confusion

Fractions Strategy: Memorization The complexity of fractions makes it more likely that students will forget that fractions represent quantities. This leads to memorization without understanding: “Find the common denominator, then add” “Flip it and multiply” “The bigger the denominator the smaller the fraction” The complexity of fractions makes it more likely that students will forget that fractions represent quantities. This leads to memorization without understanding: “Find the common denominator, then add” “Flip it and multiply” “The bigger the denominator the smaller the fraction”

Fractions: Prerequisite Understanding Solid understanding of foundational numeracy Quantity Part-whole relationships Equal groupings Reversibility Solid understanding of foundational numeracy Quantity Part-whole relationships Equal groupings Reversibility

Source: OECD PISA 2006 database Hong Kong-China Finland Korea Netherlands Liechtenstein Japan Canada Belgium Macao-China Switzerland Australia New Zealand Czech Republic Iceland Denmark France Sweden Austria Germany Ireland Slovenia United Kingdom Poland Chinese Taipei Estonia Macao-China

-- National Center for Educational Statistics, 2007 The Nation’s Report Card US Students Proficient in Math

Center for Research in Math & Science Education, Michigan State University Top Achieving Countries

Center for Research in Math & Science Education, Michigan State University 1989 NCTM Topics by Grade

Number of Topics Grade US vs Top Achieving Countries US vs Top Achieving Countries

4th Grade “There are 600 balls in a box, and 1/3 of the balls are red. How many red balls are in the box?” International Test Item

“Teachers face long lists of learning expectations to address at each grade level, with many topics repeating from year to year. Lacking clear, consistent priorities and focus, teachers stretch to find the time to present important mathematical topics effectively and in depth.” -- NCTM Curriculum Focal Points Changing Course Changing Course

‣ Focus on developing problem solving, reasoning, and critical thinking skills. ‣ Develop deep understanding, mathematical fluency, and an ability to generalize. ‣ Focus on developing problem solving, reasoning, and critical thinking skills. ‣ Develop deep understanding, mathematical fluency, and an ability to generalize. ‣ Instruction should devote “the vast majority of attention” to the most significant mathematical concepts. NCTM Recommends NCTM Recommends

Math curricula should: ‣ Be "streamlined and should emphasize a well-defined set of the most critical topics in the early grades." ‣ Emphasize "the mutually reinforcing benefits of conceptual understanding, procedural fluency, and automatic recall of facts." ‣ Teach with "adequate depth." ‣ Have an "effective, logical progression from earlier, less sophisticated topics into later, more sophisticated ones." ‣ Have teachers regularly use formative assessment. Math curricula should: ‣ Be "streamlined and should emphasize a well-defined set of the most critical topics in the early grades." ‣ Emphasize "the mutually reinforcing benefits of conceptual understanding, procedural fluency, and automatic recall of facts." ‣ Teach with "adequate depth." ‣ Have an "effective, logical progression from earlier, less sophisticated topics into later, more sophisticated ones." ‣ Have teachers regularly use formative assessment. The manner in which math is taught in the U.S. is "broken and must be fixed." National Math Panel Report National Math Panel Report

“A major goal for K-8 mathematics education should be proficiency with fractions, for such proficiency is foundational for algebra and, at the present time, seems to be severely underdeveloped.” National Math Panel Report National Math Panel Report