On Squares, Rectangles, and Square Roots - Introduction

There is some consensus in curricular guidelines, such as the National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics [10], about the necessity of enhancing the human aspect of mathematics. As a human creation, mathematics is part of our everyday life and culture, and individuals are expected to understand and appreciate it. Better knowledge of how different ideas emerged throughout the history of mathematics, and to which problems or questions these ideas gave a solution, helps to reinforce the role of mathematics in everyday life and in science and other disciplines.

Interest in knowing how history of mathematics can be used to improve the instruction and the learning of mathematics is not new. Mathematics educator Michael Fried [6] gave the following reasons why educators should consider the history of mathematics:

1. History of mathematics humanizes mathematics. It offers an enriching multicultural approach, providing students historical role-models, and linking the study of mathematics with human emotions and motivations.
2. It makes mathematics more interesting, more comprehensible and more attainable, adding more variety to teaching, decreasing students' fear and rejection of mathematics, and reinforcing the role of mathematics in society.
3. History of mathematics gives a new perspective to concepts, problems and their solutions. From the point of view of students, history provides a context for problems and ideas, suggests alternative approaches to problem-solving, and shows the relationships between ideas, definitions, and applications.

With respect to the last item, the gaining of perspective is not limited only to students, since teachers and researchers also benefit from the study of history. The historical study of a particular mathematical concept can determine a method to make an epistemological analysis of it and to identify modern students' possible difficulties. According to Guy Brousseau [2], epistemological obstacles can be detected by comparing the history of mathematical topics and today's students' mistakes in learning them. In other words, mathematics education researchers should identify obstacles encountered during the history of mathematics and compare these historical obstacles with those faced by today's learners in order to determine their epistemological character ([2], p. 42).

There are several ways to incorporate history of mathematics into teaching practice, but they follow two basic strategies. On the one hand, a teacher may introduce historical anecdotes or specific historical problems; and on the other hand, she may use a determined historical development to explain a specific technique or idea and/or to organize subject contents according to a historical scheme [6].

Frank Swetz [13] suggested the importance of allowing students to take an active part in historical discoveries: to experiment with the measurement of \(\pi\), to employ Eratosthenes' technique to measure the circumference of the earth, to use Greek ruler and compass construction, and to experience approximations of the roots of algebraic equations. Reimer and Reimer also recommended that teachers integrate the history of mathematics in different ways [12], to include:

• reading aloud mathematical stories to the class;
• having students write about mathematical history topics;
• having students perform plays, enact stories, or make videos about historical topics;
• providing experiences with manipulatives; and
• providing mathematical experiences through the arts.

Powers and square roots are one of the topics in the arithmetic section in the final level of primary education in the Spanish curriculum. At the end of this educational stage, students must be able:

• to recognize the base and exponent of a power;
• to read, write, and calculate powers of numbers;
• to connect the square and square root of a given number;
• to compute the square roots of small numbers; and
• to solve problems involving squares and square roots.

The square root is defined as the inverse operation of the multiplication of two identical factors, with textbooks for this course usually reading that "the square root of a number is another number whose square is the given number". Students have no problem in determining that the square root of \(36\) is \(6,\) because \(6^2=36,\) or that the square root of \(81\) is \(9,\) since \(9^2=81.\) The square roots of small three-digit numbers, for example \(169,\) are found by looking for the number that was multiplied by itself to get \(169.\) In this way, the student makes successive attempts, \(11^2=121,\) \(12^2=144,\) and \(13^2=169,\) to conclude that the square root of \(169\) is \(13.\)

Teaching squares roots should comprise estimation skills together with certain intuitive and innovative components. As stated in the NCTM Principles and Standards [10],

When students can connect mathematical ideas, their understanding is deeper and more lasting. They can see mathematical connections in the rich interplay among mathematical topics, in contexts that relate mathematics to other subjects, and in their own interests and experience. Through instruction that emphasizes the interrelation of mathematical ideas, students not only learn mathematics, they also learn about the utility of mathematics.

When children build with, interact with, and handle manipulative materials, they can learn mathematical concepts through significant experiences that guarantee a deeper and better internalization of the processes and concepts on which they work. Godino, Batanero, and Font [7] consider as manipulative materials physical objects taken from the environment or specifically built for the purpose, as well as graphics, concrete words, sign systems, etc. acting as expressions, explorations, or a means of calculating in mathematical work. For these authors, manipulative materials bridge the real world with mathematical objects.

The aim of this paper is to describe an educational experience carried out with a group of students in the sixth and final grade of primary education (11-12 years old) and intended to promote the learning of a method for computing square roots. Being aware that the curricular treatment given to square roots in secondary education focuses on the algorithmic aspect, we intended to provide a more meaningful learning experience for these sixth-grade students. To this end, we introduced a historical method for computing square roots, and had students use manipulative tools to facilitate the interaction of geometric and arithmetic procedures required by this method. We believe we were successful in introducing what is a complex subject at this educational stage.