Reflections: November 1998

Contents of Volume 23, No.4.

Editorial

Point of view

• Accelerating students in Junior High School, Steve Thornton 1
• Incorporating technology into the curriculum, Roger Brown 1

Reflections on high school geometry

• Historical aspects of teaching geometry relevant to Year 10, Lindsay Grimison 3
• Geometry in Years 7 to 10, Bill Pender 9

Reflections on geometrical understanding

• The SOLO model, geometrical questions and the Course Performance Descriptors, John Pegg 14

Reflections on geometrical ideas

• Gems in geometry, James Taylor 18
• Further geometrical explanations of triangles ó Antiparallels and the symmedian point, Chris Sattler 22

Reflections on teaching ideas in geometry

• The geometry of stars, Steve Thornton 27
• An open-ended geometrical question, Carol Neville 30
• Twenty cent geometry 31
• 3-Dimensional shapes 33

Reflections on technology

• Interactive geometry, Paul White and Bernice Sharp 38
• Personal technology and data analysis, Barry Kissane 40
• Creating chaotic flowers using Mathematica ó A senior investigation activity, Matt Bower 44

Reflections on senior mathematics

• Permutations without formulas or tears, Robert Yen 48

Reflections on mathematical investigations

• Paper folding to the sun, Anne Prescott 53
• Answers to questions for teachers
• Common mathematical terms, Erik Hoekstra 55
• The tan rule for triangles, Erik Hoekstra 55
• Proof that, in any Pythagorean triad, at least one term is a multiple of 5, Teresa Ashton 56
• Uncommon trigonometric functions ó Versine A and exsecant A, Teresa Ashton 57

Reviews

• Inside Mathematics 7, Reviewer: Robert Yen 58
• Instant Graphic Calculator Lessons: Pre-calculus and Calculus, Reviewer: Judy Anderson 59
• Applications to Mathematics Incorporating Calculators, Reviewer: Judy Anderson 59

The maths exchange

EDITORIAL

A recent debate on the AAMT email list yielded an interesting Point of View from Steve Thornton. Steve argues against the acceleration of students in junior high school on the basis that teachers should be aiming for deep understanding of mathematical ideas rather than a surface understanding that tends to result in moving through the curriculum much quicker than is appropriate for many students. He recommends that teachers provide a variety of interesting investigations for their talented students rather than revisiting topics in the same ways, or, moving them rapidly on to the next topic in the program. We encourage teachers to contribute their ideas for future editions of Point of View. We also include a response to the first Point of View written by Derek Ward in Reflections, 23(3), p.3. Roger Brown offers a more positive perspective on the use of graphic calculators.

This issue of Reflections focuses on aspects of teaching and learning geometry in Years 7 to 12. Grimison has written a fascinating paper on the history of geometry in the NSW curriculum. Pender describes his approach to teaching geometry in Years 7 to 10 and offers possible investigations for teachers to try. Pegg outlines a theoretical framework, or the SOLO model that was developed by Biggs and Collis, and relates it to some geometrical questions that could be used in Year 10 with the new Course Performance Descriptors. Teachers will be able to use this advice to support their efforts in determining grades for Year 10 students.

Taylor provides an overview of his 'gems' of geometry. This is a brief summary of the presentation he gave at the MANSW Annual Conference in Parramatta this year. We are grateful that he agreed to have his conference paper included in this issue of Reflections. Another paper that explores ideas about the geometry of the triangle has been submitted by Sattler.

Several authors provide useful teaching ideas related to geometry. Thornton describes activities that could be used to motivate and challenge student thinking. Neville outlines another of her open-ended investigations, and White and Sharp describe the use of the Geometer's Sketchpad to explore geometrical ideas. Permission has also been granted by the Department of Education and Training to reproduce two of my favourite activities for Years 7 and 8: Twenty Cent Geometry and 3-Dimensional Shapes, which were originally published in Curriculum Ideas for Secondary Schools -Geometry Parts 1 and 2 - SM13 and SM14 in 1986. I have used them in classrooms with a lot of success as the students are motivated and enthused by the activities.

There are also several papers that do not have a geometrical focus. Kissane describes the use of graphic calculators to explore data, and Bower investigates the use of Mathematica in the senior school. Yen provides practical advice to teachers about the teaching of permutations and combinations in Years 11 and 12. Prescott derives the solution to a well-known investigation.

Hoekstra and Ashton have both submitted responses to Jim Stamell's questions for teachers from the second issue this year (Reflections 23(2)). We encourage others to write in with questions of interest. One that received some attention recently on the AAMT email list was the origin of the pronumeral 'm' to represent the gradient of a line. Can any readers shed light on this?

The photograph on the cover was taken at Melbourne Central shopping centre and is used with permission of the centre management. This photograph was originally used on a poster produced by the Mathematical Association of Victoria to advertise their 1998 Annual Conference. I believe that it provides some interesting angles on perspective, and displays considerable architectural creativity.

Judy Anderson,Editor