Teaching and Learning Fraction Concepts and Operations

The Institute of Education Studies (IES), part of the U.S. Department of Education, has recently released a research-based 90-page report titled, "Developing Effective Fractions Instruction for Kindergarten through 8th Grade." The report begins by detailing the poor level of understanding of fraction concepts and skills among U.S. students. For example data from international and national exams have shown that "50% of 8th-graders could not order three fractions from least to greatest...[and] fewer than 30% of 17-year-olds correctly translated 0.029 as 29/1000" (p. 6). A panel of experts reviewed dozens of studies and identified five (5) recommendations for improving fraction instruction. From page 1 of the document:Build on students’ informal understanding of sharing and proportionality to develop initial fraction concepts.Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers. Use number lines as a central representational tool in teaching this and other fraction concepts from the early grades onward.Help students understand why procedures for computations with fractions make sense.Develop students’ conceptual understanding of strategies for solving ratio, rate, and proportion problems before exposing them to cross-multiplication as a procedure to use to solve such problems.Professional development programs should place a high priority on improving teachers’ understanding of fractions and of how to teach them.This list fits well with the approaches I've found successful in my own classroom (previously) and in the experiences of my credential students in their classrooms. Learning to make sense of fraction concepts requires deliberate activities that allow for sense-making to occur. Some resources to help you design such learning activities are:Have fun with these! It is always great to see students "get it" when working with fractions.

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Rote Memorization versus Conceptual Learning

A colleague teaching mathematics at a community college recently asked if I knew of research about "the effectiveness of mass practice in terms of long-term memory."  Apologies in advance if you don't have access to an academic journal database and can only view the article abstracts.

There is research about so-called "human calculators" who do mental math with large sets of figures; in every instance these folks have developed their talent by spending countless hours memorizing arithmetic facts and relationships. Seems in the most extreme cases their brains are wired to support this sort of activity.
http://rstb.royalsocietypublishing.org/content/364/1522/1385.short
However, if we are talking about whether mass memorization equates to meaningful learning for the typical person, evidence from the 1950s demonstrated this was not the case beyond the retention of simple facts:
http://www.sciencedirect.com/science/article/pii/S0022101507632996

Learning with connections is far better. Think about early school mathematics books (in the 1700s) that had page after page of formulas to memorize related to commercial transactions involving proportional relationships among different units. The time and effort it took to try to memorize these was inordinate (and drove many away from mathematics). More "modern" textbooks by the early 1900s had students learn about proportional reasoning as a general relationship that could be applied to an infinite number of specific situations - thus by learning one conceptual relationship (which could then be memorized and also re-constructed if learned with understanding), students had a much better chance of both long-term recall and correct application.

Some recent articles related to this idea demonstrate quite powerfully the advantages to conceptual learning.
http://onlinelibrary.wiley.com/doi/10.1002/ddrr.45/pdf
http://www.academicjournals.org/ERR/PDF/Pdf%202011/Jan/Narli.pdf

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