The mental number line

The mental number line is a spatial representation of numbers. Numbers are represented spatially in ascending order corresponding to their magnitude. This mental number line representation is associated with children’s numerical competencies. It is important to train this in numerical education. Within this training, it seems more promising to train the continuous spatial character of the mental number line than focusing on the directional aspect of the mental number line.

Current trainings

When looking at trainings nowadays, most refer to some type of linear format. With those it is usually possible for the children to simply count the items to get to the right answer. However, it is good to use empty number lines to improve children’s conceptualization of a certain number range and to illustrate the principles of arithmetic operations (such as addition and subtraction). By using empty number lines you can reduce the reliance on counting strategies and improve their conceptual understanding.

Computer exercises

There is more and more evidence showing that spatial-numerical games can improve numerical and arithmetic skills in children. However, all those tests were paper-pencil based. The authors argue here that computer-supported methods may be even more successful. Besides general motivational effects, they can support learning on a cognitive level by giving fast and individualized feedback, and by adaptively providing different levels of task difficulty. The benefits of these computerized trainings seem to depend on how the trainings are actually implemented. To really contribute to early childhood mathematics, the learning environment should be accompanied by effective teaching strategies. Most of the number line trainings that we have nowadays are part of a multi-componential training, usually for the prevention or intervention of mathematical learning difficulties. In evaluations these trainings seem to have positive effects on the numerical and mathematical abilities of children.

Socially interactive game trial

When it comes to interaction in computer exercises for training the mental number line, there are specific demands on interactivity and adaptivity of the learning environment. Considering these special demands, the authors developed a web platform with interactive browser games, among which it was possible to train the mental number line. The first evaluation showed that adaptivity was very important. Players who continuously had to play against players that were better than them, easily got demotivated.

Embodied interaction

Embodied interaction refers to a trend that has emerged in the area of human-computer interaction. Humans can then interact with physical objects which have been augmented with computational abilities. These systems measure user movements and/or gestures. They have a motivational appeal that can be beneficial for learning purposes. They can also be used to support more specific learning, if the task has a spatial dimension. For instance with numerical tasks, the movement of the learner may be congruent or incongruent to the direct of the internal number line representation. This can be called bodily number line training. Research indicates many positive effects, such as increases in accuracy of guessing numbers, generalization of positive effects to an improvement in counting skills, improvements in number line estimation, and other reliable transfer effects.

Conclusion

Research clearly indicates that number line trainings are an effective way of improving the numerical competencies of children. However, most evaluations were done of multicomponent trainings, and they did not investigate the effects of the individual components. Embodied number line trainings have been investigated as well, and they in turn enhance the effectiveness of regular computer-supported number line trainings. So far though they have not been able to make those trainings socially interactive and adaptive.

Future developments

Computer-supported learning environments integrate the physical reality into computer-supported math interventions. Integrating manipulatives, such as sticks or blocks, into Tangible User Interfaces (TUI) can create new possibilities for integrating tangible material with symbolic information. The tangible materials may support learning by building on the advantages of physical manipulation while avoiding limiting exploration by constraining the range of actions and possible learning opportunities. It seems that especially geometric concepts are suitable for TUI’s. So far there is no specific number line training that uses these TUI’s. One example how TUI’s can be used for number line trainings is by using tangible rods of different lengths that may be recognized by a touch sensitive surface. A program may then output the actual numerical length of the rods, and thereby create a connection between a concrete manipulative and the symbolic representation of its length.