DEC Scan Journal : May 2013
Volume 32, May 2013 27 Contents Editorial Currents Teaching & learning Research Curriculum support Share this Resource reviews learning to count in Chinese or English, between 3 and 4 years, the course of acquisition of counting began to diverge. Four-year-olds in China made very rapid progress in generalising their counting procedures once they could count to approximately 40 compared to English speakers in the United States (Figure 1). The linguistic representation of mathematical concepts in ordinary language can affect the ease of acquisition of these concepts. Chinese speaking students start school being Figure 1 The counting development of 4-year-olds in Chinese and English (based on Miller, Smith, & Zhang, 2004) able to count further and faster (Chinese counting words are all single syllables) than their English speaking counterparts. Going beyond the known Learning to reason is integral to learning mathematics. Reasoning is also a clear focus of teaching mathematics in Japan (Sawada, 1997). The creation of the Australian Curriculum in Mathematics has also brought reasoning to the fore. The curriculum seeks to ... ensure that all students benefit from access to the power of mathematical reasoning (ACARA, 2012). Indeed, reasoning is one of the four proficiencies that operate across all of the content described in the curriculum. Yet if teachers are often able to recognise mathematical reasoning when they see it, many are less confident about knowing how to develop mathematical reasoning in their students. One challenge of developing mathematical reasoning in students is that there is more than one kind of mathematical reasoning. Within the new Mathematics K–10 syllabus which incorporates the Australian Curriculum, the Stage 4 outcome related to reasoning is a student recognises and explains mathematical relationships using reasoning (MA4-3WM). As an example of how students are expected to recognise and explain mathematical relationships, I will draw on a form of deductive reasoning described as algebraic reasoning. Unfortunately, whenever the term algebra is used, most people think only of working with symbols like x. This omits a major component of the history of algebraic thinking, sometimes described as rhetorical algebra. From the time of the ancient Babylonians to the 16th century, algebraic problems and their solutions were frequently composed solely of words (Kaput, 2008). Rather than restricting my interpretation of algebra to manipulating symbols, I understand algebraic reasoning to be thinking logically about unknown quantities and the relationships between them. Algebraic reasoning Imagine sitting in a classroom where you can see the calendar for the month displayed but with some of the dates covered. A yellow strip has been placed vertically over three numbers and a red strip placed horizontally over it to form a cross (Figure 2). The teacher asks the question, Which has the larger total; the numbers covered by the yellow strip or the numbers covered by the red strip? Why? While you ponder this question, the teacher says that first she wants you to convince yourself of the answer and then to convince a friend. You may establish your answer by determining the values of the covered numbers and carrying out the additions. Figure 2 A calendar month with some numbers hidden Learning to reason is integral to learning mathematics. ... reasoning is one of the four proficiencies that operate across all of the content described in the curriculum.