The new concrete materials for mathematics

Since the time of Cuisenaire rods or before that counters and buttons students have benefitted from the use of concrete materials in their mathematical learning. The combination of strong visuals and the ability to physically manipulate groups of objects has allowed students to move from purely physical representations of number concepts to increasingly abstract representations. This pattern of first experiencing a concept in the physical world before shifting towards symbolic representations has allowed countless students to grasp the fundamentals of mathematics. Effective teachers have always been able to maximise the benefits of this pattern ensuring that their students can not only produce the right answer but have a solid understanding of the mathematics behind it. 

In the age of computers and now tablets there have been many efforts to transfer this effective learning pattern into the digital world. Skeuomorphic representations that take digital renderings of real world objects and place them onto screens for manipulation through the computer interface have had some success in this area but an additional layer of abstraction is added that purely physical manipulations lack. Shifting the physical objects into the digital world may bring new affordances to the devices used but fail to transform the learning that was already possible with concrete materials. What was needed was a relationship between the digital and the physical that would enhance the quality of learning possible. With new tools such as easily coded robots, augmented reality and 3D printing this is a possibility that is beginning to emerge. 

One interesting option is the iPad system and related Apps made available by ‘Osmo’. In this the camera on the iPad is reconfigured to capture images from the area immediately in front of the device as it sits in a stand. The software allows for virtual objects on the iPad screen to appear to interact with objects placed or drawn in front of it. One of the Apps available presents the challenge of manipulating the trajectory of a virtual ball using objects placed or drawn in the real world space in front of the device. Students learn through an iterative process of altering angles of incidence and by adding or removing objects how the trajectory of an object can be altered. The software allows for complex interactions to be simulated and tested with immediate feedback provided on the screen. Other Apps allow students to manipulate Tangrams with added benefits occurring as a result of the augmented reality. Another App bridges the gap between the physical and the abstract by allowing students to manipulate real objects while the device reveals the associated abstract representations. Osmo brings an added dimension compared to purely digital Apps that attempt to do the same thing as its physical dimension invites collaboration. Students will want to sit and play with the physical objects and though the manipulations of these experience gradually increasing levels of engagement with symbolic representations.  

Learn more about Osmo

Sphero is another interesting way to take mathematics out of the digital world and allow it to blend with the physical. Sphero is a small spherical robot that can be controlled directly using a phone or tablet, can be programmed or can be used as a controller or interactive object with augmented reality games. Beyond the fun that comes from using Sphero like a remote controlled toy is the potential to use it to explore concepts within mathematics. This process is likely to begin with lessons that require students to programme their Sphero to complete a maze. Constructed on the floor with tape, cardboard or with a mixture of made and found obstacles mazes allow students to explore aspects of measurement of length, area, angles, time and speed. Unlike purely digital solutions that target these concepts Sphero brings a real connection between the code and the movement of the robot. Students engaging with Sphero will take measurements in the real world with traditional tools such as metre rules and protractors (preferably oversized ones), use that information in the digital code environment and then make adjustments based on the path taken by the robot. Again success is supported through an iterative approach with immediate and natural feedback. These feedback loops where the students are quickly shown what works and what does not are judgment free and ensure continued engagement with the problem and with each failed attempt revealing something new about the task while allowing the students to laugh at their robots misadventures. That students are learning to code while they are developing their understanding of mathematical concepts is further bonus. Beyond the maze activities are options for an exploration of how the robot may be programmed to complete artistic pieces that combine its movement and ability to change colours into robotic ballets that could include collaborations with other robots.  

Learn more about SPHERO

You can’t mention links between digital worlds and physical reality without mentioning 3D printing. For explorations in STEAM the increasingly affordable 3D printer brings the opportunity to design objects in a physical world and then print a real model that the students can hold. So many mathematical concepts from mass, volume, capacity, scale, ratio, and 3D shape can be explored in this way and links across key learning areas can be readily explored. For design thinking tasks the potential to have a physical prototype of an idea brings a tangible dimension that reveals ideas in new ways showing how imaginings in the virtual world translate into the real world. Increased accuracy of 3D printers is allowing for objects with great precision which means it is possible to print parts of machines with moving and interlocking cogs. Software is available now that will show how these moving parts will interact in a virtual world thus enabling testing of designs in this space prior to printing.  

In each case here there is a linking of learning in a digital space with the manipulation of creation of real world objects. instead of moving further away from the use of concrete objects blended environments suggest that mathematics teaching may be most successful when a mix of digital and physical is enabled. It may not be time to throw away those Cuisenaire rods and oversized set squares just yet, they may have a new part to play in the emerging terrain of mathematical learning.  


By Nigel Coutts