By Paul Drijvers
These days, algebra schooling is topic to around the world scrutiny. various evaluations on its targets, methods and achievements are on the middle of debates between lecturers, educators, researchers and selection makers. What may still the instructing of algebra in secondary university arithmetic appear like? may still it concentrate on procedural talents or on algebraic perception? may still it rigidity perform or combine know-how? can we require formal proofs and notations, or do casual representations suffice? Is algebra in class an summary topic, or does it take its relevance from software in (daily lifestyles) contexts? What should still secondary institution algebra schooling that prepares for better schooling perform within the twenty-first century appear like? This ebook addresses those questions, and goals to notify in-service and destiny academics, arithmetic educators and researchers on fresh insights within the area, and on particular themes and issues resembling the old improvement of algebra, the position of efficient perform, and algebra in technology and engineering specifically. The authors, all affiliated with the Freudenthal Institute for technological know-how and arithmetic schooling within the Netherlands, percentage a typical philosophy, which acts as a ? occasionally approximately invisible ? spine for the general view on algebra schooling: the idea of reasonable arithmetic schooling. From this aspect of departure, diversified views are selected to explain the possibilities and pitfalls of cutting-edge and tomorrow's algebra schooling. Inspiring examples and reflections illustrate present perform and discover the unknown way forward for algebra schooling to correctly meet scholars' wishes.
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Extra info for Secondary Algebra Education
Example text
Probably Galileo’s most widely quoted statement is: This [the book of nature] is written in the language of mathematics and the main characters are triangles, circles and other geometric figures, without which it would be impossible for people to understand a single word. v Galileo claimed especially that geometric objects are necessary for us in order to find our way in the universe. This is a philosophical point of view that outstandingly exemplifies the 17th century; Galileo’s scritto in lingua matematica is a gripping way to express this view, but the one-sided emphasis on the aspect of language in the quotation disregards the importance of the geometric objects that Galileo also refers to: triangles and circles.
The solution is therefore built up under Hau. First (1+ 1--2- + 1--4- ) is written individually, directly to the left of the ‘1’. Then this is doubled (see to the left of the 2), and doubled again (left of the 4). 1--7- of ( 1--- + 1--- +1) 4 2 is also determined, which is 1--4- ; see the fourth line. The rows marked with arrows (compare the corresponding slashes in the original) give 9 for the ‘heap’ together, and for the number of times that ( 1--4- + 1--2- +1) is multiplied, 1 + 4 + 7--7- is used temporarily; in the heap, which is still 1 too little to make the required 10.
In this system, the solution is achieved by doubling and then adding up the results. It is actually very similar to a long division problem. Giving the name of ‘Hau’ to the right-hand column can also be interpreted as ‘algebra’, but the specific numerical computations are still the main part of the text. 30 ALGEBRA FROM AHMES TO APPLET The example of Ahmes places little or no emphasis on the general aspects of the method. It is certainly not yet algebra, in the modern sense of a systematic approach to a problem that is independent of the specific numbers.