Comparing the use of the empty number line in England and the Netherlands

Abstract

Efforts to meet the needs of children's learning in arithmetic has led to an increased emphasis on the teaching of mental calculation strategies in England. This has included the adoption of didactical tools such as the empty number line (ENL) that was developed as part of the realistic mathematics movement in the Netherlands. It has been claimed that the English use of the ENL differs from that of the Dutch. In this study curriculum guidance materials are examined to investigate these differences. Two examples of teaching sequences are provided to illustrate them further. It is proposed that the different uses stem from different pedagogical principles. The implications of the English use are discussed in relation to the presentation of mental calculation strategies in an algorithmic way.

Middle Grade Students’ Evoked Concept Images of Number Line Models and Their Calculation Strategies with Integers Using These Models

“…The number line enables students to visualize the position of numbers relative to each other, helps teachers demonstrate calculations, and enables students to develop visual strategies (Bramald, 2000;Heeffer, 2011;Murphy, 2011). One of the pioneering reviews of the number line in literature was made by Ernest (1985) who summarized the ways of using the number line as follows: (1) a number line is a classroom practice which teachers use while teaching the addition and subtraction of single-digit whole numbers, (2) it is seen in written objectives involving addition and subtraction of whole numbers, and (3) it is used while making computations.…”

“…Gonsalves and Krawec (2014) stressed that a deep understanding of operations is necessary to represent operations on the number line in an accurate way. In accord with this, many researchers address the use of number line for the calculation of whole number operations as well as for the construction of arithmetical strategies with whole numbers (Bramald, 2000;Klein, Beishuizen, & Treffers, 1998;Murphy, 2011;Ryan & Williams, 2007). For example, Ryan and Williams (2007) discussed the number line as a useful tool to support arithmetic solutions of whole number problems.…”

“…Number line models are intertwined with real-life contexts for integers (Akyuz, Stephan, & Dixon, 2012;Gravemeijer, 1994;Stephan & Akyuz, 2013). The research points out that comprehension of integers by students through number line models and making sense of real-life contexts of integers from a mathematical perspective are separate issues (Bobis & Bobis, 2005;Gallardo & Romero, 1999;Murphy, 2011;Van den Heuvel-Panhuizen, 2008;Vlassis, 2004). That often causes students to make sense of integers through giving meaning to signs of operations and numbers and number line models independently.…”

Thinkpiece: Making space for mathematics learning to happen in group work: Is this really possible?

“…Foot/End Notes: These should be avoided where possible; the journal preference is for footnotes rather than endnotes. My research in primary mathematics education in England focused on the teaching of calculation strategies following the introduction of the English numeracy strategies (Murphy, 2004(Murphy, , 2011a. My recent arrival in New Zealand as a lecturer and researcher in primary mathematics has given me a fresh perspective in mathematics education, particularly in children's learning in numeracy.…”

“…These two publications, Anthony and Walshaw (2007) and the New Zealand professional development booklets (Ministry of Education, 2008a) seem to present two main approaches to teaching: one related to direct instruction, and the other related to discussion and collaboration or dialogic pedagogy. My research into teaching numeracy has led me to review philosophical perspectives that underlie different pedagogies in primary mathematics (Murphy, 2011a(Murphy, , 2012 and these two pedagogies would seem to be underpinned by contrasting structuralist and non-structuralist perspectives.…”