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In order for students of science, technology, engineering and mathematics to have a proper algebraic foundation, the secondary curriculum must go beyond memorizing results and technicalities. Algebra is a powerful tool that can be wielded only if its user has a sophisticated appreciation of its value as a language, its structure and its role in proving results. In this chapter, we look at an example that is accessible to material in the early high school syllabus in which students are encouraged to deepen their grasp of the subject.

2001:7. Suppose that x ≥ 1 and that \(x = \lfloor x\rfloor +\{ x\}\), where ⌊x⌋ is the greatest integer not exceeding x and the fractional part {x} satisfies 0 ≤ { x} < 1. Define $$\displaystyle{f(x) = \frac{\sqrt{\lfloor x\rfloor } + \sqrt{\{x\}}} {\sqrt{x}}.}$$

2001:2. Let O = (0, 0) and Q = (1, 0). Find the point P on the line with equation \(y = x + 1\) for which the angle OPQ is a maximum.

The integer points on the line and the edges between them can be coloured \( 1 --(3) --2 --(1) --3 --(2) --1 \) and so on, where the edge colouring is in parentheses. Form a plane by stacking these lines unit distance apart, making sure that each vertex has a different coloured vertex above and below it; use colours 4 and 5 judiciously to colour the vertical edges. Now go to three dimensions; stack up planar lattices and struts unit distance apart, colouring each with the colours 1, 2, 3, 4, 5, while making sure that vertically adjacent vertices have separate colours, and use the colours 6 and 7 for vertical struts. Continue on.

Except for the first contest, for which 3 h was allotted, the time allowed for each contest was \(3\frac{1} {2}\) h.The chapter in which the solution appears is given in parentheses at the beginning of the problem.

Yehuda Rav’s inspiring paper “Why do we prove theorems?” published in Philosophia Mathematica (1999; 7: 5–41) has interesting implications for mathematics education. We examine Rav’s central ideas on proof – that proofs convey important elements of mathematics such as strategies and methods, that it is “proofs rather than theorems that are the bearers of mathematical knowledge” and thus that proofs should be the primary focus of mathematical interest – and then discuss their significance for mathematics education in general and for the teaching of proof in particular.

Zazkis, R., & Campbell, S.R. (Eds.). (2006). Number Theory in Mathematics Education: Perspectives and Prospects. Mahwah, NJ: Lawrence Erlbaum. ISBN 0–8058–5407‐X (cloth); 0–8058–5408–8 (paper)

From time to time, the International Commission on Mathematical Instruction (ICMI) mounts studies to investigate in depth and detail particular fields of interest in mathematics education. This paper is the Discussion Document of the forthcoming ICMI Study 16, Challenging Mathematics in and beyond the Classroom.

In this Paper we generalize some recent results concerning sequences satisfying recursions essentially of the type where r is a positive integern ranges over the integers, and ∊ is equal to 1.0 or -1. By means of a map in R which keeps invariant certain surfaces, we study the dynamic of the solutions of such recursions and identity their periodicities.

Inevitably, in such discussion groups, the time is too short to handle topics to the depth that they deserve. It is hoped that these notes will provide some hint of the discussions in the tertiary algebra subgroup and serve as a basis for further delibertions among participants and others.

(1992). Commutativity of Polynomials. The College Mathematics Journal: Vol. 23, No. 5, pp. 386-395.

When the school curriculum is subdi vided into short units and the class moves quickly from one topic to the next without penetrating very deeply into any of them, students are in danger of coming away without any sense of purpose or con text. Although facts and techniques may be covered, their significance and power may be unrealized, so that students quickly forget them. For this reason, introducing mathemati cal situations requiring more lengthy con sideration is worthwhile. Their analysis will pique students' interest in the curriculum and incidentally furnish practice in basic skills. The purpose of this article is to give two examples of such in-depth study that are suitable for students who are beginning the study of algebra.

This chapter gives many examples of challenging problems, categorized according to the settings in which they occur, both in and beyond the classroom. Some challenges arise as extensions of normal classroom practice; other challenges, some of very recent origin, become popular among the general population, while still others are created especially for contests for different populations of students. In some cases, a detailed discussion of their origins and uses is provided. It should become clear to the reader the extent to which they betoken the vitality and the creativity of the mathematical enterprise and show how much of a ‘‘human endeavor’’ mathematics is. The final section provides a summary of the place of context and content in the creation and use of challenges. This chapter focuses on challenges given in the form of individual problems; challenges that occur in extended investigations are mentioned briefly in the final section and will be considered in later chapters.

The first chapter presented a situation that led to pairs of integers (x, y) that satisfied equations of the form x 2 - 2y 2 = k for some constant k. One of the reasons for the popularity of Pell’s equation as a topic for mathematical investigation is the fact that many natural questions that one might ask about integers lead to a quadratic equation in two variables, which in turn can be cast as a Pell’s equation. In this chapter we will present a selection of such problems for you to sample.

In this chapter, we consider analogues of Pell’s equation of higher degree, particularly the fourth. Throughout, n is a positive integer and c is an integer which has a real nth root 6 of the same sign. Define $${g_{c}}({x_{1}},{x_{2}},....,{x_{n}}) \equiv N({x_{1}} + {x_{2}}\theta + {x_{3}}{\theta ^{2}} + \cdots + {x_{n}}{\theta ^{{n - 1}}}) \equiv \prod\limits_{{i = 0}}^{n} {({x_{1}} + {x_{2}}({\xi ^{i}}\theta ) + \cdots + {x_{n}}{{({\xi ^{i}}\theta )}^{{n - 1}}})} $$ where \(\varsigma = \cos \frac{{2\pi }}{n} + i\sin \frac{{2\pi }}{n} \) is a primitive nth root of unity. The analogue of Pell’s equation that we want to examine is $${g_{c}}({x_{1}},{x_{2}}...,{x_{n}}) = 1. $$

When d is a positive nonsquare integer, Section 4.1 assures us that there is a solution for x 2 - dy 2 = 1. However, we do not have much guidance on how actually to lay our hands on it. In this chapter we will develop algorithms that will lead to a solution.

In Section 3.2, we saw how, using various tricks, solutions in rational x and y of x 2 — dy 2 = 1 could be obtained from two solutions of an equation x 2 - dy 2 = k. Sometimes, the rational numbers turned out to be integers. The chances of this happening would apparently improve with the number of solutions of x 2 - dy 2 = k for a particular k. This suggests that it might be useful to look for values of k for which there are a lot of solutions. This strategy succeeds spectacularly; there are infinitely many solutions for suitable k.

The material in this chapter was developed by Jeffrey Higham in 1993, when he was an undergraduate at the University of Toronto. We have already seen that with respect to the multiplication $$(x,y)*(u,v) = (xu + dyv,xv + yu) $$ defined for pairs of integers, the solutions of Pell’s equation x 2 - dy 2 = 1 can be obtained by taking “powers” of an elementary solution. We will explore this structure when we consider instead pairs (x, y) for which x 2 - dy 2 differs from 1 by a multiple of some fixed modulus m.

In looking for a higher-degree version of Pell’s equation, a natural choice is the equation $${x^{3}} - {\text{ }}d{y^{3}} = {\text{ }}k, $$ where d is a noncubic integer. However, it turns out that the solutions of such equations are not very numerous, nor do they exhibit the nice structure found in the quadratic case. This chapter will begin with an investigation of this type before treating a better analogue that admits a theory comparable to the quadratic version. This is the equation: $${x^{3}} + c{y^{3}} + {c^{2}}{z^{3}} - 3cxyz{\text{ }} = {\text{ }}k,$$ where c is any integer other than a perfect cube and k is an integer. In this chapter we will see how this equation comes about and examine its theory. It will turn out, as in the quadratic case, that there is a fundamental solution; however, this solution is not so neatly obtained and some ad hoc methods are needed.

To arouse interest in Pell’s equation and introduce some of the ideas that will be important in its study, we will examine the question of the irrationality of the square root of 2. This has its roots in Greek mathematics and can be looked at from the standpoint of arithmetic, geometry, or analysis.

In an unpublished note Golomb proposed a family of "strange" recursions of metafibonacci type, parametrized by $k$. Previously we showed that contrary to Golomb's conjecture, for each $k$ there are many increasing solutions, and an explicit construction for multiple solutions was displayed. By reformulating our solution approach using matrix dynamics, we extend these results to a characterization of the asymptotic behaviour of all solutions of the Golomb recursion. This matrix dynamics perspective is also used to construct what we believe is the first example of a "nontrivial" nonincreasing solution, that is, one that is not eventually increasing.