## Intention

Our intention has been to write a comprehensive series of eighty short concept based chapters designed to engage and motivate teachers and students in higher-level secondary mathematics courses corresponding to the Specialist and Methods courses in the Australian National Curriculum.

We did not intend the series to be an abbreviated skills based textbook. Rather, in what we believe is an effective pedagogical approach, we tried to place each piece within the context of the narrative of human endeavor. We imagine that Recorde would have approved of our aim in the series to build deep understandings in a first-principles conceptual way – building each story from an initial enquiry and working through to the formulation of ideas and statements.

Mathematics in schools may at times and in some places, be presented solely as a utilitarian activity: we study it because it is useful and, if useful, then worth knowing about. However, when it is broken down into topics and day-to-day lessons there is a risk that students may see the subject as a series of unrelated theoretical constructions completely detached from their own experience and devoid of real meaning. It can become algorithmically driven; replete with processes, formulae, substitutions and solutions. How many students have ever used the

It is true that Mathematics underpins our scientific understandings and technological advances but if we can get students to see that mathematics is many other things as well, then perhaps the relevance of the subject and students’ connectedness to it will be enhanced.

We would prefer to convey the idea that Mathematics is a way of thinking – logical, sequential and well defined. It is a celebration of culture and human endeavor – full of intrigue, inspiration, creativity and discovery. Mathematics is an art – rich in form, symmetry, and beauty, just as nature tends to present itself. Mathematics is an adventure – a journey into the unknown.

A teacher’s success in revealing these other attributes hinges largely on the way the subject is delivered. To this end, the Whetstones that we have put together have been written with an emphasis on the mathematical thinking and the triumph of discovery rather than simply on utilitarian value.

As an example, think for a moment about the teaching of the

We did not intend the series to be an abbreviated skills based textbook. Rather, in what we believe is an effective pedagogical approach, we tried to place each piece within the context of the narrative of human endeavor. We imagine that Recorde would have approved of our aim in the series to build deep understandings in a first-principles conceptual way – building each story from an initial enquiry and working through to the formulation of ideas and statements.

Mathematics in schools may at times and in some places, be presented solely as a utilitarian activity: we study it because it is useful and, if useful, then worth knowing about. However, when it is broken down into topics and day-to-day lessons there is a risk that students may see the subject as a series of unrelated theoretical constructions completely detached from their own experience and devoid of real meaning. It can become algorithmically driven; replete with processes, formulae, substitutions and solutions. How many students have ever used the

*sine rule*after school? How many have used Pythagoras’ theorem? When have they ever needed to distinguish a rational number from an irrational one?It is true that Mathematics underpins our scientific understandings and technological advances but if we can get students to see that mathematics is many other things as well, then perhaps the relevance of the subject and students’ connectedness to it will be enhanced.

We would prefer to convey the idea that Mathematics is a way of thinking – logical, sequential and well defined. It is a celebration of culture and human endeavor – full of intrigue, inspiration, creativity and discovery. Mathematics is an art – rich in form, symmetry, and beauty, just as nature tends to present itself. Mathematics is an adventure – a journey into the unknown.

A teacher’s success in revealing these other attributes hinges largely on the way the subject is delivered. To this end, the Whetstones that we have put together have been written with an emphasis on the mathematical thinking and the triumph of discovery rather than simply on utilitarian value.

As an example, think for a moment about the teaching of the

*sine rule*. We might draw a triangle, label it and write down the equation.If we have enough time we might even prove the result. After that, we might set up a specific example, with three things known and one to find – perhaps an angle or a side, and then set a few practice questions to answer.

But, a

But, a

*Whetstone*might take a different tack. It might remind students of the old adage from Euclid that*things equal to the same thing are equal to each other*; so that the sine rule should be writtenAnd in the quantity,

The constant d ties everything together. No matter what triangle we create, each side and its opposite angle have a very special relationship to each other. The symmetry is beautiful – the side

Investigating the sine rule in this way becomes a rich activity – it is learning at a deeper level, and the thing about deep understandings is that they are not just remembered for a while, but understood for a lifetime. They are not just algorithms for solving problems, but realizations of the connectedness, symmetry and form of the world. Understandings are not just useful formulae, but they are statements of discovery, and as such are energizing and motivating. This sentiment characterizes the collection of four-page

*d,*lies the hidden magic of the result.The constant d ties everything together. No matter what triangle we create, each side and its opposite angle have a very special relationship to each other. The symmetry is beautiful – the side

*a*is different from the side*b*, their opposite angles*A*and*B*are different as well, but the ratio of*a*to the sine of*A*is precisely the same as the ratio of*b*to the sine of*B*, and these ratios are equal to a constant*d*. And how is related to any specific triangle? Is its value significant? Is each different triangle associated with a different*d*? How large or small can*d*get?Investigating the sine rule in this way becomes a rich activity – it is learning at a deeper level, and the thing about deep understandings is that they are not just remembered for a while, but understood for a lifetime. They are not just algorithms for solving problems, but realizations of the connectedness, symmetry and form of the world. Understandings are not just useful formulae, but they are statements of discovery, and as such are energizing and motivating. This sentiment characterizes the collection of four-page

*Whetstones*that comprise this book.