# References

Below we give an annotated list of books that you may find helpful in giving you further ideas for problem solving lessons. If you have others that you think we should add to the list please E-mail us the details.

Carl B. Boyer, A History of Mathematics, John Wiley & Son, New York, Second Edition, 1991.

This rather fat book gives a readable account of the history of mathematics from the year dot up to the last century. However, its Twentieth Century helpings are limited, probably due to the fact that much of it is hard to explain and it may not yet be clear who are the important people and results from this recent time in history.

Nevertheless you’ll find something on all the people in mathematics that you’ve heard about (Pythagoras, Newton, Euler, Gauss, and so on) and probably a lot more that you haven’t. Although the emphasis is on Western Europe, China and India share a chapter and there is a chapter on the Arabic contribution to mathematics.

Hill, T. (1987). *Work it out: Strategies for Problem Solving
in Maths*. Oxford University Press: Australia.

Work it outprovides teachers of middle and upper primary students with ten strategies for problem solving in maths. These strategies are introduced with sample problems and are followed by reproducible worksheets for year levels 3 to 6.

Skinner, P. (1990). *What's your problem?: Posing and Solving
Mathematical Problems* *in Junior Classes.* Thomas Nelson: Australia.

"What's your problem?" describes the problem solving mathematics programme that author Penny Skinner developed with a class of junior school children over a 2 ½ year period. Teachers are provided with clear and detailed information on how they can set up similar programmes in their own classes. Chapters cover: how to organise time, content and evaluation, how to make a start with problems, writing and posing problems, sharing problems and developing problem solving strategies. The emphasis in the programme described is upon the presentation of mathematical concepts and skills through problems, many of which should be generated by the children themselves.

Stacey, K. & Groves, S. (1985). *Strategies for Problem
Solving: Lesson Plans for Developing Mathematical Thinking*. Victoria College
Press: Victoria, Australia.

Strategies for Problem Solving was written for junior secondary mathematics teachers although none of the activities required mathematical content beyond that of the primary school. The book provides teachers with a practical synthesis of problems, strategies and theory, in a format that is suitable for immediate classroom use. It contains detailed lesson plans, teacher support materials and reproducible student worksheets for about 40 lessons, each designed to teach important problem solving skills and strategies.

Stacey, K. & Southwell, B. (1991). *Teacher Tactics for
Problem Solving. *Curriculum Corporation: Melbourne.

Teacher Tactics for Problem Solving contains 38 problems supported by information that helps the teacher and the student get the most out of the problem. Each problem is presented in the following format:

Skills to be developed – lists the problem solving skills and strategies that children might use to solve the problem. Materials required Mathematics required Problem Adaptation for younger students. Stuck? – These are questions that teachers could pose for students who are stuck. Extensions – suggestions for other avenues to explore. Solution and notes – a brief outline of one or more solutions is given.

Ministry of Education*. Teaching Mathematical
Problem Solving in Mathematics: Years 1 – 8.* Wellington: Learning Media,
1999.

This publication consists of a small booklet and a CD ROM. The 32-page booklet contains a number of useful hints on problem solving ranging from what it is to what its advantages and disadvantages are. It also talks about one-off problem solving lessons as well as units of work using problem solving. Along the way it makes suggestions about strategies, scaffolding children and assessment.

The CD ROM gives a number of lessons and units of work to support the material of the booklet. These cover Levels 1 to 4 of the curriculum. There is a facility for you to create your own lessons. In addition, the CD ROM contains assessment tasks and certificate blanks that you can use for children in your class.

Holton, Derek, Anthony Neyland, Jim Neyland and
Bronwen Thomas. *Teaching Problem Solving*. Chichester: Kingsham Press, 1999.

This book provides an introduction to the teaching of problem solving for primary and junior secondary teachers. The various ideas relating to problem solving are introduced by specific problems of varying difficulty. Problems in the text are generally solved in a ‘chatty’ way. One chapter is devoted to case studies where problems are solved on one side of the page and discussed on the other. In some way this is supposed to represent the doing of the problem (on the left) and the thinking about the problem (on the right).

At the end of each chapter there are a number of questions to reinforce the material of that chapter. The book ends with six sample lessons.

The book should be useful for teachers and pre-service teachers alike.

Holton, Derek and Charles Lovitt. *Lighting
Mathematical Fires*. Carlton, Victoria: Curriculum Corporation, 1998.

This is essentially a book written for junior secondary school but some of the ideas and problems will be relevant to the intermediate school.

The book is divided into four sections. The first deals with general problem solving matters and gives some thought to teaching problem solving. The second section contains ten problems (along with master sheets that can be copied for students) to illustrate the use of different strategies. The third section has another ten problems, which give students a chance to practice their strategies, and allows teachers the opportunity to extend the list of known strategies. The final section suggests ways that problem solving problems can be generated from old, known situations and new, invented ones. The section suggests that all teachers are capable of producing their own problems for their own students.

Curran, John (Ed.). *Problem Challenge: 5-Year
Competition Book 1991-1995*. Dunedin: Department of Mathematics and Statistics,
University of Otago.

The

Problem Challengeis a competition for intermediate school students (as well as talented mathematical students in lower years) that has been going since 1991. Students attempt 5 questions in 30 minutes 5 times in the year. Book tokens are awarded to the top 1% of students, the top 10% get certificates of excellence, the top (roughly) 25% get a certificate of merit, and the remainder of the students get a certificate of participation.The book starts with some tips on problem solving going over standard strategies with worked examples. It then gives the 25 questions from each of the first five years of the competition. An answer section follows these as do full solutions for each question. The last section of the book provides the statistics for the

Problem Challengefor the years 1991 to 1995.As a result the book is a useful store of good problems that can be used in a variety of ways. They can be used as one-off questions to extend the class (especially its more able members); as the basis for class lessons; as practice for next year’s

Problem Challengeand so on.The book is an excellent and valuable resource and can be obtained through the University of Otago. For any enquiries about the

Problem Challengecontact John Curran in the Department of Mathematics and Statistics. His email address is jcurran@maths.otago.ac.nz.

### Other Web-sites

http://www.japanese-online.com/math.

As the front page says:

"The story problems provided here are translated from Japan’s Junior High School math placement test. This test is given to 12 year olds and each section of the full test consists of 225 story problems. Students are given a time limit for each problem ranging from 1 to 5 minutes. If completed within the time provided, the 225 story problems require over 8 hours to complete."The story problems are logic-based and consist of about 20 different types of story problems. The point of this site is to begin providing quality math content based on world standards."

This is an American site consisting of 20 multiple choice word problems. You might find it a little surprising that the Japanese students manage to solve any of these problems in that time. To help you there are hints as well as explanations of the answers.

A typical example follows:

Jenny wanted to purchase 2 dozen pencils and a pen. These items cost $8.45 and she did not have enough money. So she decided to purchase 8 fewer pencils and paid $6.05. How much was a pen?The problems might be useful at Level 4 but only exceptional students should be given the time limit. You might be able to work on the problems to make them more accessible for your students. But it is interesting just to look and see what students in Japan are expected to be able to do.

http://www.nrich.camb.org.uk: details to come