# Trains of thought

A David Acheson's 1089 and All That takes a user-friendly approach to a variety of maths, providing stimulation to look further, with little teasers and great cartoons. The chapter "Great mistakes" opens with a story: "An astronomer, a physicist and a mathematician were on a train journey together in Scotland. Glancing from the window, they observed a black sheep in the middle of the field. 'How interesting!' said the astronomer. 'All Scottish sheep are black!'

"The physicist, rather startled, said: 'Surely you mean some Scottish sheep are black?'

"But the mathematician viewed even this as a bit rash. 'I think what you both mean,' he said, 'is that there is at least one sheep in Scotland which is black on at least one side.'" The chapter then goes on to describe Malfatti's problem: how to pack three circles into a triangle so that they cover the largest total possible area and don't overlap. Malfatti's solution was to choose the circles in such a way that each one touches two sides of the triangle and both the other circles.

As Acheson explains, this was thought to be correct until, in 1967, Michael Goldberg demonstrated that it never is. One of the circles must always touch all three sides. Was he right? Is there more? This investigation provides lots of practice in working out areas of circles in triangles. The book provides a fuller explanation and solutions found by various mathematicians, as well as other activities that would stimulate an interest to find out more.

At www.jesus.ox.ac.uk dacheson you can find simulations of planetary motion, convergence of infinite series, and more - well worth a visit!

1089 and All That by David Acheson, published 2002 by Oxford University Press, pound;12.99 Q Where can I find a comprehensive list of common misconceptions that students have in mathematics?

A Below are listed four websites where you can access ways in which students have or might develop misconceptions in mathematics. The first contains an article on misconceptions in mathematics research by Professor J Van Horn. It might be interesting for you to try the "Students and Professor" problem quoted below with your colleagues and students - they might be encouraged to know that 35per cent of engineering students who tried it gave an incorrect solution.

This example would make a good lesson starter for tackling word problems. After discussing the solutions, students could create their own word problems for each other to try.

"Write an equation using the variables 'S' and 'P' to represent the following statement: 'There are six times as many students as teachers at a certain school. Use 'S' for the number of students and 'P' for the number of professors." (The solution to this problem is at the end of the column.) The incorrect solution is most likely caused by the reader translating the sentence as it stands, instead of thinking of the sense of the equation - a simple substitution of values into the equation would show that. The translation of word problems is where many students have difficulty, mainly because they try to tackle all the words at once. Time spent on teaching them how to translate words into symbols and mathematical sentences is invaluable in their progress towards mathematical application. In the initial stages, the substitution of values to help understand the process should not be undervalued.

The first website includes the article mentioned above and refers to a number of articles and books about misconceptions in maths.

www.penpages.psu.edupenpages_reference28507285073220.html

www.mathsyear2000.orgresourcesmisconceptions index.shtml

The next site contains mistakes common among undergraduates but is worth a visit.

http:library.trinity.wa.edu.ausubjectsmathsmismaths.htm

www.math.vanderbilt.eduschectexcommerrs

Pete Griffin, curriculum adviser for mathematics in Devon, suggests Errors amp; Misconceptions in Maths at Key Stage 2 by Mike Spooner, published by David Fulton at pound;12.

"Students and professor" solution Many incorrectly write 6S = P instead of the correct S = 6P.

Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses. Email your questions to Mathagony Aunt at teacher@tes.co.ukOr write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX

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