Academic journal article Australian Mathematics Teacher

Constructing with Non-Standard Bricks

Academic journal article Australian Mathematics Teacher

Constructing with Non-Standard Bricks

Article excerpt

Play is our brain's favorite way of learning. (Diane Ackerman)

Introduction

The necessity of using inquiry-based learning (IBL) was recently recommended by studies and reports made for the European Commission (see Wallberg-Henriksson, Hemmo, Csermely, Rocard, Jorde & Lenzen, 2006). Several European projects are devoted to the widespread use of IBL methods (see the ProCoNet group at http://proconet.ph-freiburg.de). Moreover, the effects of using IBL are studied worldwide (see Laursen, Hassi, Kogan, Hunter & Weston, 2011). In the framework of the Seventh Framework Program (FP7) project PRIMAS (1), a series of piloting activities were organized in Romania in order to test, adapt and develop inquiry-based teaching materials. Most of these piloting actions were organised by local professional communities with the purpose of creating real feedback for the project and for gathering professional experience in implementing inquiry-based pedagogies in mathematics and science education. The main aim of this paper is to present an activity where students were formulating the problems. Teachers were only creating the milieu (Brousseau, 1997) and facilitating the work. As a second step, the accumulated experience related to this activity was used in a professional development (PD) course organised by the Babes-Bolyai University in the framework of the PRIMAS project.

During the activities, students worked in small groups; each group had two types of pieces as shown in Figure 1a and they had to construct and plan patterns and objects. The patterns and objects were not specified: each group decided what to construct. One basic rule was fixed: all groups had to present all successful and unsuccessful attempts. Moreover, for all the constructions, a careful analysis was necessary. It is worth mentioning that the number of pieces was relatively small for each group (12-36), but in constructing the patterns and objects they had to deal with an unlimited number of pieces.

Constructing patterns and 3D objects

Before our activities, we organised a series of playing activities with the set of Marble and Profi cubes (see www.happycube.com; Andras, Sipos & Soos, 2011). The first object constructed was the cube, while in the plane most of the groups constructed rectangular configurations. After these they constructed rectangular parallelepipeds (see Figure 1), square pipes, curved square pipes and various other shapes (see Figure 3).

During the analysis of the plane configurations, most groups observed that if we want to cover a certain planar region, we do not have very many options. In fact, there exists only one possible way to fit the pieces and this arrangement generates a covering of the plane (see Figure 6). The formal proof of this fact was not requested, but the students formulated an inductive argument, which emphasised that a configuration formed by pieces is repeated.

Similar arguments were given for the construction of square pipes, moreover, students found a connection between the covering of the plane and the construction of the pipe. They observed that if we unfold the pipe, we obtain a strip from the covering of the plane. This observation assures that a pipe with arbitrary length can be constructed (see Figure 5 and Figure 4).

Using an additional argument students showed that any curved pipe can be constructed (the ratios of straight sections can not be arbitrary, but the direction can be changed at any step, see Figure 3). The determination of the number of needed pieces for such a construction was also carried out by the students.

The first surprise appeared when students wanted to construct a 9 X 9 X 9 cube (which should contain 2 X 2 pieces on each side) and they failed.

After a few attempts they formulated that it is impossible to construct a 9 x 9 x 9 cube (without holes). Some groups constructed a 9 x 9 x 9 cube where two small (1 x 1 x 1) cubes from the opposite corners were missing, so they also conjectured that this is the best approximation of the 9 x 9 x 9 cube with the given pieces. …

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