By Dina Tirosh

What arithmetic is entailed in realizing to behave in a second? Is tacit, rhetorical wisdom major in arithmetic schooling? What is the position of intuitive types in figuring out, studying and instructing arithmetic? Are there changes among trouble-free and complex mathematical pondering? Why cannot scholars turn out? What are the features of lecturers' methods of figuring out? This ebook specializes in quite a few different types of wisdom which are major for studying and instructing arithmetic. the 1st half defines, discusses and contrasts mental, philosophical and didactical concerns concerning a variety of sorts of wisdom interested in the training of arithmetic. the second one half describes principles approximately different types of mathematical wisdom which are very important for lecturers to grasp and methods of imposing such principles in preservice and in-service schooling. The chapters supply a large evaluate of present puzzling over arithmetic studying and educating that is of curiosity for researchers in arithmetic schooling and arithmetic educators. subject matters coated comprise the function of instinct in arithmetic studying and instructing, the expansion from straightforward to complicated mathematical pondering, the importance of genres and rhetoric for the training of arithmetic and the characterization of academics' methods of understanding.

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**Extra resources for Forms of Mathematical Knowledge - Learning and Teaching with Understanding**

**Example text**

1989, ‘Not all preconceptions are misconceptions: Finding ‘anchoring conceptions’ for grounding instruction on students’ intuitions’, International Journal for Science Education 11, 554–565. : 1967, The Philosophical Works (Vol. 1), (translated by E. S. Haldane and G. R. T. Ross), The University Press, Cambridge, MA. DiSessa, A. : 1988, Knowledge in pieces, in G. Forman and P. B. ), Constructivism in Computer Age, Erlbaum, Hillsdale, NJ, pp. 49–70. R. : 1975, The Intuitive Sources of Probabilistic Thinking in Children, D.

Printed in the Netherlands. 52 DINA TIROSH AND RUTH STAVY twice as much water as the other, they claimed that ‘the more water– the warmer’ (Erickson, 1979; Stavy and Berkovitz, 1980). This response is often interpreted as an alternative conception of temperature. Another example relates to children’s conceptions of angle. Noss (1987) presented children with two identical angles, one of which had ‘longer arms’ than the other. They found that many children between the ages of ten and fifteen argued that ‘the angle with the longer arm is bigger’.

Such conceptions cover a wide range of subject areas. Most of this research has been content-specific and aimed for detailed descriptions of particular alternative concepts. Yet, there is evidence that students tend to respond inconsistently to tasks related to the very same mathematical or scientific concepts (Clough and Driver, 1986; Nunes, Schliemann and Carraher, 1993; Tirosh, 1990). This constitutes a challenge to the alternative conception paradigm. Through our work in mathematics and science education, we have observed that students react in a similar way to a wide variety of conceptually nonrelated problems which share some external, common features.