br Acknowledgements This work was supported
Acknowledgements This work was supported by the European Committee Fifth Framework Project QLK5-1999-01360. We are grateful to Dr. T.T. Hoang for the provision of plasmids pEX18Tc, pPS858 and pFLP2, Prof. B.L. Wanner for providing us with the plasmid pKD4 and related strains, and Prof. P. Soucaille for the gift of E. coli MG1655 ΔtpiA::FRT mutant strain. We wish to thank K. Weber and Dr. H. Bloecker for their help in sequencing. We would also like to thank Dr. M. Schobert for valuable discussion and Dr. H.-W. Ma for helping with making the alverine tree.
Introduction One of the most useful representations of number is a number line. The number line is important as it can be used for connecting decimals, fractions, and other types of numbers and for developing a sense of relative size. It is an especially important representation because, unlike many other representations and models used in teaching, the number line plays an important role in mathematics to the most advanced levels. It is used for measuring scales and Cartesian axes, horizontal and vertical, as well as embodying , the abstract set of real numbers. The number line is also important for learning because of its epistemic fidelity (Stacey, Helme, Archer, & Condon, 2001) towards characteristics of real numbers such as density; the property that in between two unequal numbers there is another one. A discrete model, such as multi-base arithmetic blocks, cannot faithfully present this property. Hence, the number line is important because it is both a didactical model and also a part of higher mathematics. This paper reports a detailed examination of the thinking of a sample of pre-service primary teachers when placing decimal numbers on number lines. We provide evidence that many of them exhibit a twisted geography of the negative side of the number line. We identify two major misconceptions that cause high error rates on apparently basic tasks. There are important consequences both for school education and teacher education. The main introductory work on the number line at school tends to be limited to representing positive and negative integers, and negative fractions and decimals are often not explicitly treated. It is also the case that instruction about calculation with fractions and decimals concentrates on positive numbers, and that negative decimals and fractions receive little attention. As will be noted below, our own teaching has fallen into this trap and we have observed it in many textbooks.
Results In this section, we describe the students’ behavior then in Section 5 we provide an explanation of the thinking behind the behavior.
Exploring the thinking behind the errors
Discussion and implications The study found a low incidence of positive decimal misconceptions in this population, which is consistent with other reports such as Stacey, Helme, Steinle, et al. (2001) on pre-service teachers. Difficulties in scaling, applicable to placing both positive and negative decimals on the number line, were evident in some wrong answers (e.g., in Fig. 6), however, the study has found further evidence for documented difficulties with positive decimals. It supports an unexpected finding from work on positive decimals by Steinle and Stacey (2003) that decimals with zero integer part (e.g., 0.35) are conceptually harder than decimals with a non-zero integer part, and so would benefit from special treatment in teaching. Misconceptions with negative decimals are also evident when students deal with negative integers. For example, a personal communication from Cecilia Kilhamn (2 February 2010) reports analysis of video of a Swedish teacher helping students calculating the age at death of Emperor Augustus who was born in the year 63BC (i.e., 63 BCE) and died in 14AD (i.e., 14 CE). The teacher expects students to know the answer is 63+14, but wants them to write 14−(−63). Eight students ask the teacher for help. Of these eight students, two students want to use 37 in the calculation. Hans asks: “But didn’t he live 37 years before Christ was born?” and with the same thinking, Victor comments that Chi structure is 37 years from −63 to 0. Fig. 13 shows, firstly, a correct number line (time line) where there are 37 years from 1963 to the turn of the century and, secondly, the incorrect mental model used by these boys when thinking about crossing the 0 barrier (1940 and −40 have been placed for clarification). These students’ comments are consistent with both the misconceptions above but, rather than misplacing intervals of length 1 or 2, they have misplaced the interval of 100 years. Further evidence would be required to decide whether they have translated the positive interval [0,100] to the negative side (misconception TP1 or TP2), or whether they have imagined a separate negative number ray [−0, …, −40, … −63, … −100) with −100 near 0 AD.