A dictionnary of the Human Orrery

Linking language and gestures to create meaning: a dictionary of the "human Orrery" (extract from the article published on the BUP in 2020 – in French)

The use of the human Orrery is based on the link between sensory experience and the construction of abstract concepts.

This article describes the association between gestures and movements on the human Orrery and abstract kinematic concepts. This association may be referred to as a first step towards a dictionary of the human Orrery. The ultimate aim is to encourage "interaction between different sensory modalities and different signs" that will enable the learners' "perceptions and associated mathematical thoughts" to "acquire a theoretical dimension".

Terms and gestures associated with duration

To move the pupils around the Human Orrery, but also to move the pawns, the game master (in this case the teacher, but it could also be a pupil) claps his hands regularly. This action sets the "musical" rhythm and the kinematic terms associated with it need to be well defined. The essential aspect of the discourse associated with the gestures must be the distinction between moment and duration.

A clap: At the moment when the script supervisor claps his hands, a sound is heard. We refer to the perception of the clapping hands (for the handler), the hearing of the sound (for the actors) and the associated moment (for everyone) as a "clap".

The duration between two claps or the frequency of claps: The duration between two "claps" is constant (depending on the regularity of the master of the game, a metronome can be used...). We have noticed that both the pupils and the teacher often talk about "frequency", sometimes implicitly, i.e. the number of claps heard in a second (or rather over a constant duration, not explicitly defined). It may be preferable to talk only about the duration between two claps and to define the concept of frequency, which has a more complex relationship with speed, at a later stage. Note here that the duration of a clap has no meaning, because a clap corresponds to an instant.

Two other durations are frequently used: the duration of a step and the duration of a revolution around the Sun.

The duration of a step: this duration is imposed as equal to the time between two claps, whereas the length of a step is left free. We'll come back to this difference in our analysis of the embodiment of speed.

The duration of a revolution: this is a special kind of time that can cause problems.  Knowing the duration of a revolution for a planet fixes its angular velocity and also its linear velocity, because the distance travelled is fixed. In the case of the movement of two bodies (Earth and Mercury, for example), if they took the same amount of time to make one revolution, they would have the same angular velocity but different linear velocities. Although angular velocity is not part of the secondary school curriculum, we had the feeling that this notion was intuitive for the students and that it was implicit in their discourse. 

Duration scale: The time taken by the student embodying the Earth to make one revolution links the "real" duration (one year) to the duration of our model (measured by a stopwatch, for example). The duration of the model can be modified by changing the frequency of the claps, whereas the real duration cannot be modified. By calculating this scale, we can, for example, measure the actual time taken for Mercury to make one revolution, i.e. the duration of a 'Mercurian' year.

                Terms and gestures associated with travel, position and distance

There are three types of movement on the planet. Once again, the association of these movements with positions and distances must be clearly specified. The distinction between position and distance (or length) is fundamental to the concept of speed. The definitions below also show the links between distance and time that can be made for each movement.

A step: "A step" corresponds to the movement made during the time between two claps. This movement is not necessarily from one point on the sunbed to another. Thus, the initial and final positions of a "step" are not imposed by the design of the sun: the duration of this movement is imposed, but not its length.

The points/discs of the Human Orrery: Each disc naturally corresponds to a position, as the students (or the pieces) will move from one point to another. The exact position is the center of the disc. This movement can take several steps, but each point on the Human Orrery will be reached at a given moment. The distance between the points is observed directly on the Human Orrery and will therefore be covered in one or more steps: the duration of the journey from one point to another is not imposed, but its length is fixed.

The perimeter of an orbit: This is a specific distance on the planet. Again, the time taken to cover it is not fixed, but its length is.

Distance scale: This scale is imposed by the drawing of the Human Orrery and cannot therefore be modified. The Earth's semi-major axis can be determined geometrically on the drawing. This length measured on the Human Orrery corresponds to an "Astronomical Unit" for the real Solar System.

Reading these associations, it becomes clear that displacements (distance and position) do not have the same status as durations (and instants) on the planet. These differences need to be explained in words or actions to avoid confusion. The interplay between these quantities can then be used to work fruitfully on speed.   

How is speed embodied?

The length covered or measured is embodied by vision (points on the Human Orrery) and by kinaesthetic action (moving the body or pawns), in an active way. Duration is embodied passively by hearing the clap. How is speed embodied? 

Before answering this question, we would like to stress that the expression "the speed of the claps" has no meaning in the context of the Human Orrery. Claps should only be associated with an instant (for a clap) and a duration (between two claps). Note that it might be interesting to look, in an independent session, at whether the hands necessarily move faster when the time between two claps decreases (for example).

The concept of speed is neither visible (like distance) nor audible (like duration). A priori, there is no sense with which to "measure" speed. As mentioned above, speed must be understood as a link between distance and duration. However, everyone can feel and perceive whether they are travelling at a significant speed or not, whether they are slowing down or accelerating. We think that these feelings are mainly comparisons of speed. This can be done at a fixed distance. For example, in a car, if the houses in the landscape, of fixed width, pass in front of our eyes in less time, we know that the speed of the car has increased. This comparison can also be made for a fixed distance. The speeds of two cars can be perceived by looking at the distance they cover over the same short period of time. It is worth remembering here that any speed measured is an average speed measured over a period of time which depends, of course, on the observation tool (eye, radar or even the Doppler effect with a very short, but non-zero, time!). Common reasoning then assumes that the "speed" does not vary over this time interval, becoming equal to an instantaneous speed. Instantaneous speed is more intuitive than average speed ("theoretical speed", the result of a mathematical operation), which is introduced later in the school and social context. What's more, the concept of speed (seen as "instantaneous") does not necessarily imply an upstream link with the concepts of distance and time interval for all pupils.

To explain the way in which speed is embodied on the Human Orrery in relation to the notions of distance and duration, it is interesting to break down the movement into three beats and around 4 positions (A, B, C and D). Let's emphasise that these positions are not necessarily points on the Human Orrery (for example, the points on Encke's orbit would be located at positions A and D, since it takes 3 steps to go from one point to the other).

Instant 1	Between instant 1 and instant 2	Instant 2	Between instant 2 and instant 3	Instant 3
The right foot is placed in position A and the left foot in position B. A clap sounds.	The right foot rises from position A and moves towards position C. The whole body has thus covered a certain distance between the two positions.	The right foot is placed in position C and the left foot in position B.	The left foot rises from position B and moves forward to position D	The right foot is placed in position C and the left foot in position D.

This proposed choreography can be modified according to ideas and desires, in particular in conjunction with sports’ teachers, while remaining attentive to two aspects that seem important to us. Firstly, the distinction between moment/duration and position/movement must be clear to the teacher in order to introduce these concepts to the pupils without confusion. Secondly, the movement must be continuous: avoid stopping between two claps and taking a quick step at the moment of the clap. This is necessary so that the notion of instantaneous speed makes sense in this movement. This (ideal) objective is not easy to achieve and the pupils are often very concentrated on their step.