Recommended Readings
Hyperlinks to papers on the internet have been provided where they are available, but many of the readings are contained within books and journals.
Estimation and children’s concept of rational number size
Estimation and children’s concept of rational number size - Behr, M. J., Post, T. R., & Wachsmuth, I. (1986).
In H. Schoen & M. Zweng (Eds.), Estimation and mental computation: 1986 NCTM Yearbook (pp. 103-111). Reston, VA: National Council of Teachers of Mathematics.
Estimation and Children's Concept of Rational Number Size (http://cehd.umn.edu/rationalnumberproject/86_1.html)
This article emphasises the importance of conceptualizing a fraction as a single entity and then being able to attribute size to this entity. It discusses two estimating tasks (comparing fractions and “construct a sum”) that were being used in the Rational Number Project to assess student understanding of the size of fractions.
Order and equivalence of rational numbers
Order and equivalence of rational numbers: A clinical teaching experiment - Behr, M., Wachsmuth, I., Post, T., & Lesh, R. (1984).
Journal for Research in Mathematics Education, 15(5), 323-341.
Order and equivalence of rational numbers (http://cehd.umn.edu/rationalnumberproject/84_2.html)
This article describes in detail the strategies inferred from students explanations when asked to compare fraction pairs of three types: same numerators, same denominators, and different numerators and denominators. These interviews occurred during an 18-week teaching experiment with fourth-grade students.
Fractions as division: The forgotten notion
Fractions as division: The forgotten notion - Clarke, D. (2006).
Australian Primary Mathematics Classroom, 11(3), 4-10.
Through a game involving sharing chocolate, the author explores an often neglected construct of fractions, that is, fractions as division or quotient. It draws our attention to the important understanding that fractions can be represented as the result of division such that 2/3 is 2÷3 or two shared between three.
Year six fraction understanding: A part of the whole story
Year six fraction understanding: A part of the whole story - Clarke, D. M., Roche, A., & Mitchell, A. (2007).
In J. Watson & K. Beswick (Eds.), Proceedings of the 30th annual conference of the Mathematics Education Research Group of Australasia (Vol. 1, pp.
207-216). Hobart: MERGA.
This article describes the results from the use of a one-on-one interview with 323 grade six students on a range of tasks assessing student understanding about fractions, but also includes an in-depth discussion on student strategies when comparing fractions.
The Value of making mistakes
The Value of making mistakes - Eggleton, P. J., & Moldavan, C. C. (2001).
Mathematics Teaching in the Middle School, 7(1), 42-47.
This article emphasises the importance of students “owning” their errors and using these mistakes as a catalyst for learning. The author advocates that teachers reassess their views on student errors, so that errors can be viewed in a positive light and as a part of a problem solving process. It describes an activity whereby students must line-up cards containing, fractions, decimals and percents in order from smallest to largest.
Organizing diversity in early fraction thinking
Organizing diversity in early fraction thinking - Empson, S. (2002).
In B. Litwiller, & G. Bright (Eds.), Making sense of fractions, ratios, and proportions (2002 Yearbook) (pp. 29-40)Reston, VA: National Council of Teachers of Mathematics.
This article provides a framework for making sense of how young students solve and discuss equal sharing problems. The author supports the focus on student generated strategies and representations to provide opportunities to discuss different mathematical aspects of fractions.
Teaching fractions and ratios for understanding
Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (2nd Ed.) - Lamon, S. J. (2006). Mahwah, NJ: Lawrence Erlbaum.
More: In-depth discussion of the reasoning activities in “Teaching fractions and ratios for understanding” (2nd Ed.) - Lamon, S. J. (2006). Mahwah, NJ: Lawrence Erlbaum.
These two books provide a practical explanation of the key ideas in proportional reasoning. The first book gives the background to the topic, and a large number of problems which the reader is invited to try personally and with students. The second book provides worked solutions to each of the problems.
Knowing and teaching elementary mathematics in China and the United States
Knowing and teaching elementary mathematics in China and the United States (pp. 55-83) - Ma, L. (1999).
Mahwah, NJ: Lawrence Erlbaum.
This author conducted a study that compares the mathematical understanding of U.S. and Chinese primary teachers. One task in the study challenged teachers to solve a expression involving division of fractions and to provide a meaningful representation for the mathematical sentence. What results was an interesting insight into the mathematical knowledge of the teachers involved and invites readers to question their own.
What mathematics do adults really do in adult life?
What mathematics do adults really do in adult life? Northcote, M., & McIntosh, A. (1999).
Australian Primary Mathematics Classroom, 4(1), 19-21.
This article reports a study where 200 adults were asked to make notes on all calculations they did in a 24-hour period. Two of the important findings were that over 60% of the time, only an estimate was required, and that over 84% of all calculations involved were mental, with only 11% involving pen and paper.
Understanding decimals
Understanding decimals - Moloney, K., & Stacey, K. (1996).
The Australian Mathematics Teacher, 52(1). 4-8.
This article describes some of the ways in which students think about decimal notation, how this changes over time and provides a simple test that teachers can use to assess the understanding of their students.
What do you do when you don’t know what to do?
What do you do when you don’t know what to do? Pool, P. (2003).
Mathematics Teacher, 182, 42-44.
This article describes a fraction problem that was posed to able thirteen year olds and some of their approaches to solving this novel and c hallenging task.
Teaching computational estimation: Concepts and strategies
Teaching computational estimation: Concepts and strategies - Reys, B. J. (1986).
In H. Schoen & M. Zweng (Eds.), Estimation and mental computation: 1986 NCTM Yearbook (pp. 103-111). Reston, VA: National Council of Teachers of Mathematics.
This article highlights the importance of developing number concepts (number sense) through activities involving estimation, with some particular examples with fractions and decimals.
Longer is larger-Or is it?
Longer is larger-Or is it? Roche, A. (2005).
Australian Primary Mathematics Classroom, 10(3), 11-16.
This article outlines some student misconceptions about decimal fractions, examines some tasks and suggests teaching implications and strategies.
Multiplication with fractions: A Piagetian, constructivist approach
Multiplication with fractions: A Piagetian, constructivist approach - Warrington, M. A., & Kamii, C. (1998).
Mathematics Teaching in the Middle School, 3(5), 339-343.
This article describes the author’s experiences with her class of year five and six students when teaching division with fractions without teaching the algorithm of “invert and multiply”.
Proportional reasoning: Some rational thoughts
Proportional reasoning: Some rational thoughts - Wright, V. (2005).
In J. Mousley, L. Bragg, & C. Campbell (Eds.), Mathematics: Celebrating achievement (pp. 363-375). Melbourne: Mathematical Association of Victoria.
This article explores the five constructs of rational numbers (part-whole comparison, measures, operators, quotients and rates and ratios) in the context of proportional reasoning.