On Squares, Rectangles, and Square Roots - Discussion and conclusions

In this work we have presented a classroom experience that exemplifies a feasible use of mathematics history in primary education. The didactical design introduces the students to the procedure known as Kai Fang for the calculation of the square root and is supported using manipulative materials. The method, used in ancient China, combines arithmetical and geometrical practices in such a way that it provides a wider network of meanings for the mathematical objects put into play than those of the usual algorithm. Therefore, our experience supports the line of thought of other authors [5], who claim that both teachers and students need to learn that mathematics is much more than the isolated knowledge of a series of calculation procedures. Along these lines, the history of mathematics can serve as an instrument to expand this reductionist view and to influence attitudes and beliefs about mathematics.

As noted in the preceding section, the types of student errors we detected are as follows:

• When students express the result, they identify the decomposition of the radicand with the root (see Figure 8a), or they do not recognize the radical symbol and write that the root of the number is equal to the number itself.
• Arithmetic or expression errors; equalities are interwoven in the operations carried out to decompose the area.
• The decomposition does not correspond to the sum of the areas of the squares and rectangles (see Figure 8b).
• The student does not complete the square well (see Figure 9).

a) Mistake when expressing the root of\(\sqrt{196}\).	b) Radicand decomposition does not match the sum of the areas in the calculation of \(\sqrt{256}\).

Figure 8. Arithmetical mistakes.

It is clear from the results that the students use different additive decompositions on their way to finding the perfect square and the requested root. Although it would be necessary to have a larger sample of participants and a greater diversity of numbers on which to calculate their roots to know for sure, the results in Table 1 indicate that, in our case, students had no more difficulties in breaking down the number 576 compared to 196.

The student fails to decompose the square to calculate \(\sqrt{196}\).	The student fails to decompose the square to calculate \(\sqrt{256}\).

Figure 9. Geometrical mistakes.

The time devoted to the activity was short: the explanation, examples and resolution all were done in one hour. Even with this short time, the results were satisfactory from the cognitive and affective points of view and strengthen our interest in tasks of this typology. It would be interesting to be able to devote more sessions to reinforce or remember the Kai Fang technique, even for the computation of non-exact square roots, by means of new rectangles.

It should be noted that the required physical manipulation to graphically represent the arithmetic expressions as areas of flat figures (squares and rectangles in this case) enriches the didactic sequence. Duval [4] points out the need to consider changes of semiotic system in the tasks to be carried out and the knowledge to be developed by the students, because in this process one gains in understanding about the mathematical object in question.

The work developed in the decompositions involves a sequence of contextualized arithmetical expressions, which also allows students to deepen their understanding of the equals sign "as a symbol of mathematical equivalence" [8]. This aspect is often neglected in primary education and therefore it is frequently identified as a source of difficulties when passing from arithmetic thinking to algebraic thinking. In addition, the proposed task previews work with polynomial identities in secondary education, providing a series of physical actions that can later be evoked to facilitate, for example, the identification of the square of a sum.

Furthermore, we consider this task to be amenable to and enhanced by the use of a virtual manipulative, performed, for example, with GeoGebra. The procedures the students used are illustrated by the two applets are provided at the start of this article. We repeat these applets below.

The first applet allows users to compute two-digit integer square roots (from 10 to 99) of integers between 100 and 9801.

The second applet allows users to compute three-digit and some four-digit square roots.