«Vectors in Climbing Anne Birgitte Fyhn1 University of Tromsø, Norway Abstract: In this article, the work on mesospace embodiments of mathematics is ...»
TMME, vol7, nos.2&3, p.295
Vectors in Climbing
Anne Birgitte Fyhn1
University of Tromsø, Norway
Abstract: In this article, the work on mesospace embodiments of mathematics is further
developed by exploring the teaching and learning of vector concepts through climbing
activities. The relevance and connection between climbing and vector algebra notions is
illustrated via embedded digital videos.
Keywords: climbing and mathematics; digital media; egocentric vs allocentric
representations; embodied mathematics; flow; mathematisation; mathematical archaeology;
mathematics and physical education; meso space; meso space representations; reflective research practice; vectors; teaching and learning geometry Introduction The teaching of vectors is not an easy task for mathematics teachers. It is difficult to illustrate the concepts of vector calculus exclusively by means of blackboard and chalk (Perjési, 2003), and it also is difficult to succeed in the teaching of vectors (Poynter & Tall, 2005). The literature on the teaching and learning of vectors is very scant in mathematics in comparison to physics education. The dearth of studies addressing student difficulties in vector concepts seems troubling given the basis it forms for vector Calculus, applied analysis and other areas of mathematics. Fyhn (2007, 2010) used children’s experiences with physical activities and body movement as a basis for the teaching of angles. In this previous work the focus was on angles in a climbing context. One idea behind the work with angles in a climbing context for primary school students was to lay the groundwork for further work with vectors in upper secondary school (ibid.). The Norwegian curriculum introduces vectors as part of the mathematics syllabus in the second grade of upper secondary school.
A climbing video can introduce vectors without a blackboard and chalk, and such a video might aid in students’ understanding. In addition, a video has the possibilities of freezing the picture and drawing vectors on it, in order to guide the watcher’s focus. This paper presents and analyses the climbing-and-vectors-video “Vectors in Climbing” found at http://ndla.no/nb/node/46170?from_fag=56.
-Click link to activate video stream- The actors in this video are two skilled climbers, Birgit and Eirik, who are second year students at upper secondary school. Both had chosen theoretical mathematics as a subject at 1 Førsteamanuensis Matematikkdidaktikk Lærerut. og pedagogikk Universitetet i Tromsø firstname.lastname@example.org The Montana Mathematics Enthusiast, ISSN 1551-3440, Vol. 7, nos.2&3, pp.295-306 2010©Montana Council of Teachers of Mathematics & Information Age Publishing Fyhn school. To research whether this video may be a useful contribution to the teaching of vectors, the following question was posed How are vectors introduced in the video “Vectors in climbing”?
The choice of context Students should learn mathematics by developing and using mathematical concepts and tools in day-to-day situations which make sense to them (Van den Heuvel-Panhuizen, 2003). The fantasy world of fairytales as well as the formal world of mathematics might function as contexts for mathematical problems. The point is that the students have to experience the context as “real” in their own worlds of thoughts and imagination (ibid.).
Climbing is an activity that can be carried out by most young people given the opportunity and basic instruction. Several places in Norway have indoor climbing walls at a reasonable distance from the upper secondary schools. Many classes include students who climb regularly as a recreational activity. Students with some climbing experiences are familiar with the climbing context, and they will be able to imagine what goes on in the video. Those with no climbing experiences will to some extent be able to imagine what goes on. Most people have experienced climbing trees or rocks as children.
The concept of ‘flow’ was introduced by Csikszentmihalyi (2000). Flow is the holistic feeling one gets when one acts with complete involvement. Humans seek flow for its own sake, and not as a means for achieving something else. Professional athletes have referred to the notion of flow during moments of peak performance and described it as a feeling that transcends duality.
Achievement of a goal is important to mark one’s performance but is not in itself satisfying. What keeps one going is the experience of acting outside the parameters of worry and boredom: the experience of flow. (ibid. s. 38) One quality of flow experiences is that they reveal clear and unambiguous feedback to a person’s acting. Climbing, chess and mathematics are examples of flow activities (ibid.). The climbing video offers vectors as an integrated part of an activity which is experienced as exciting and fun by some young people. In the video Birgit and Eirik claim, “… in climbing you always have to … have fun!” Bodily geometry2 Human mathematics is based upon experiences with our bodies in the world, and therefore the only mathematics we are able to know is the mathematics that our bodies and brains allow us to know (Lakoff & Núñez, 2000). When someone presents you with an idea, you need the appropriate brain mechanism to be in place for you to (hopefully) understand it, and then learn it or reject it. Consequently, “mathematics is fundamentally a human enterprise arising from basic human activities” (ibid. p. 351). This perspective matches Freudenthal (1973), who claimed that geometry is grasping space. “Space” here means the space in which the child lives, breathes and moves. The goal of teaching of geometry is that the students are able to live, breath and move better in the space (ibid.).
Bodily geometry is geometry teaching that builds upon the students’ own experiences with their complete bodies present in a three dimensional world. The goal of bodily geometry is a) that the students are able to understand the outcomes of their own bodily actions, and b) that the students are able to use geometry as a tool for living and moving better in particular contexts. The two central goals of the video are i) that students are able to use climbing as a resource for their understanding of vectors, and ii) that students may use their knowledge about vectors to analyse what they do and then improve some of their movements. The last goal mainly concerns students who are climbers.
Different conceptions of space According to Lakoff and Núñez (2000) space is conceptualized in two different ways through the history of mankind. We cannot avoid the first one, the ‘naturally continuous space’; “It arises because we have a body and a brain and we function in the everyday world. It is unconscious and automatic” (ibid., s. 265). Climbing takes place in this naturally continuous space where coordinates and axes do not exist. Descartes’ metaphor, ‘numbers-as-points-on-a –line’ lead to the metaphor ‘space-as-a-set-of-points’, which is ubiquitous in contemporary mathematics. Even though it takes special training to think in terms of the set-of-points metaphor, it is taken for granted throughout contemporary mathematics (ibid.): One aim in the Norwegian mathematics syllabus LK-06 was “to visualize vectors in the plane, both geometrically as arrows and analytically in co-ordinate form.” (KD, 2006, p. 3), and one more aim was “to calculate and analyze lengths and angles to determine the parallelity and orthogonality by combining arithmetical rules for vectors” (ibid.). The video concerns vectors in the naturally continuous space and no calculations take place. A large part of the video focuses on how vectors may be added. A basic understanding of this is necessary before the students are able to succeed in vector calculus.
Berthelot and Salin (1998) subdivided space into three different categories with respect to size; microspace which corresponds to grasping relations, mesospace which corresponds to spatial experiences from daily life situations, and macrospace which corresponds to the mountains, the unknown city and rural spaces. Poynter and Tall (2005a; 2005b) used the term physical activity if the students move an object they hold in their hand, even if they are sitting on a chair performing a micro space activity. This is not bodily geometry; bodily geometry takes place in meso space. The vector video presents bodily geometry, because the actors move their own bodies in meso space.
The mathematical abstraction ‘line segments’ are connected with ‘long objects’ via the phenomenon of the rigid body. And a rigid body remains congruent with itself when displaced (Freudenthal, 1983). A climber is familiar with her or his own limbs and body. One point of climbing is not to fall off the wall; how to place the body and how to make it rigid.
Freudenthal further claimed I am pretty sure that rigidity is experienced at an earlier stage of development than length and that length and invariance of length are constituted from rigidity rather than the other way around. Rigidity is a property that covers all dimensions while length requires objects where one dimension is privileged or stressed. (ibid., p. 13) In bodily geometry a rigid body is an important part of ‘length’; a long arm is able to reach longer than a short arm. And if you stand on your toes you are able to reach a little further.
Fyhn Climbing: a natural alternation between egocentric and allocentric frames of reference Berthoz (2000) referred to ‘personal space’, ‘extra personal space’ and ‘far space’, where personal space in principle is located within the limits of a person’s own body. According to Berthoz (ibid.) the brain uses two different frames of reference for representing the position of objects. The relationships between objects in a room can be encoded either ‘egocentrically’, by relating everything to yourself, or an ‘allocentric’ way, related to a frame of reference that is external to your body. Only primates and humans are genuinely capable of allocentric encoding. Children first relate space to their own bodies and the ability of allocentric encoding appears later (ibid.). Moreover, “allocentric encoding is constant with respect to a person’s own movement; so it is well suited to internal mental simulation of displacements” (ibid., p. 100).
When you are trying to ascend a passage of a climbing route, you encode the actual passage egocentrically within your personal space. But when you stand below a climbing route considering whether or how to ascend it, you exercise allocentric encoding in extra personal space by considering how the route’s different elements and your body relate to each other.
The climber Fredrik explains how this takes place http://www.uvett.uit.no/iplu/fyhn/fredrik/fredrik_forklarer.html
-Click link to activate video streamClimbing can offer students good possibilities for moving back and forth between egocentric and allocentric representations. While one climber is struggling with a problem, other fellow climbers are often keen on trying to solve the problem themselves. Through their allocentric encoding they may imagine how to solve the problem. Next, they make use of egocentric coding in their attempts to solve the problem.
Mathematical archaeology One idea of the video was to start with an activity which many students find exciting and fascinating, and then search for some of the mathematics within the activity. This is done through mathematical archaeology (Fyhn, 2010). A mathematical archaeology is an educational activity where mathematics is recognized and named. This involves being aware that some activities are in fact mathematics (Skovsmose, 1994). “An aim of a mathematical archaeology is to make explicit the actual use of mathematics hidden in the social structures and routines” (ibid., p. 95). The term ‘archaeology’ refers to a systematically ‘un-earthing’ (Torkildsen, 2006) of something hidden. In this case vectors are un-earthed from three climbing situations. In Fyhn (2010), a detailed description of how angles concepts were unearthed was given. This idea is now extended to the realm of vectors.
The idea was to offer a more thorough presentation of vectors in one particular context, instead of making a presentation of all aspects of vectors. So the intention was not to start with the learning goals in the syllabus, and then search for some teaching that will fulfill these demands. Climbing is an activity, which requires full concentration. Every move concerns problem solving; how do you position your body so that your hands and feet do not loose the grip of the holds? Climbing does not necessarily concern bodily geometry, because most climbers do not reflect on any mathematics while climbing. By performing mathematical archaeology on some climbing situations, bodily geometry can be the focus.
TMME, vol7, nos.2&3, p.299 Vectors might be a useful tool for explaining what goes on in some climbing situations, but it is not given that these situations will constitute a proper basis for the teaching of vectors. This is one limitation of this approach.
Intuitive and formal understanding Fischbein (1994) claimed, that mathematics should be considered from two points of view; as formal deductive knowledge as found in high-level textbooks, or as a human activity. He pointed out that the ideal of mathematics, as a logically structured body of knowledge, does not exclude the necessity to consider mathematics as a creative process. Mathematics is a human activity, which is invented by human beings.