Nancy Mack published a series of papers (Mack 1990, 1995, 2001) were she explores the possibility of using informal knowledge as a basis to teach fraction operations with reasoning (as opposed to emphasizing procedures).
Firstly, she focused on addition and subtraction (Mack 1990, 1995) and then on multiplication (Mack, 2001).
In the later, after collecting data from individual instructional session with average ability students, she concluded that it is possible to rely on the informal knowledge of "partitioning into equal parts" to promote reasoning about questions related to multiplication of fractions. Her analysis showed that, with proper instruction, the students were able to apply their informal knowledge in a host of different questions formulated in therms of "real life problems".
However, when it comes to addition and subtraction, the author concluded that the informal knowledge suffered the interference of rote procedures and integer number bias and the students were not able to capitalize on it to solve the questions as they were with multiplication of fractions. She stresses that the students were able to solve questions formulated in terms of "real life problems" not not when formulated symbolically. This phenomena was observed even after direct instruction.
In my research, I am focusing only on addition and subtraction of fraction and, so far, it seems that the extensive use of visual representations (rectangular area model for fractions) are enabling the use not only to solve symbolic and contextualized questions, but also explain verbally how and why they solved the questions the way they did.
What I expect by the end of my research project is to collect evidences that the visual representations are effective as basis for students to reason about addition and subtraction of fractions and a in depth description of the sequence of lessons used to achieve that goal.
Mack, N. K. (1990). Learning Fractions with Understanding: Building on Informal Knowledge. Journal for Research in Mathematics Education, 21(1).
Mack, N. K. (1995). Confounding whole-number and fraction concepts when building an informal knowledge. Journal for Research in Mathematics Education, 26(5).
Mack, N. K. (2001). Building on Informal Knowledge through Instruction in a Complex Content Domain: Partitioning, Units, and Understanding Multiplication of Fractions. Journal for Research in Mathematics Education, 32(3).
Date: 11 Feb 2016
Author: Leonardo Barichello