Let's Talk About Math against Symbol-Shock. Ronald Greber. Socratic psychology consulting. . Abstract. About 1/3 of young adults ...
Greber, R. (2015). Let's talk about math against symbol-shock.
Let’s Talk About Math against Symbol-Shock
Ronald Greber Socratic psychology consulting
Abstract About 1/3 of young adults between the ages of 16 and 20 are not literate enough in either reading advanced texts nor in reading comprehensive mathematical expressions with operators. As one of several explanations for this kind of ‘illiteracy’, we take into account that the languages of mathematics, physics, chemistry, etc. all contain unfamiliar symbols, or possible ‘shock inducing elements’, which we propose leads to a phenomenon called ‘symbol-shock’. As well as having other symptoms ‘symbol-shock’ may provoke symptoms associated with frightening or freezing behavior. Shock symbols are defined as ‘non-meaningful gaps’ which cannot be interpreted or filled with any meaning. From our quasi-experimental study of consulting young adults, we propose several heuristics for prevention of symbol shock. We propose a set of self-management instructions to establish a cognitive and emotional meta-system to minimize ‘symbol-shock’.
Keywords: symbol-shock, trauma, literacy, mathematics, meta-cognition
Observations Literacy of mathematical terms with variables and function symbols is often not sufficient to succeed in an exam scenario. Among many factors influencing the performance in mathematical comprehension and problem solving (interest in mathematics, individual attitude to science, learning behavior, learning environment, etc.) we focused our study in particular on the confrontation of the symbolic language of mathematics. Previous studies by Kettler published in 1991 and 1998, showed that the combinations of algebraic mathematical symbols and operators may cause a cognitive and emotional behavior called ‘symbol shock’. This behavioral response of being shocked appears in a similar manner to psychological testing with the Rorschach inkblot test where the sudden change of black and white testing cards to cards with colored inkblots reportedly caused disruption in behavior came to be known by Rorschach as ‘color-shock’. In his early studies by Rorschach (Bohm, 1996) the subjects stopped interpreting the inkblots for several seconds which is interpreted as a dishabituation response in cognitive psychology.
Symbol-Shock as a Metacognition For a basic understanding of the ‘symbol-shock’ phenomenon we refer to the research of metacognition in cognitive psychology (Flavell, 1979): ‘Metacognitive experiences are any conscious cognitive or affective experiences that accompany and pertain to any intellectual enterprise. An example would be the sudden feeling that you do not understand something another person just said.’ (p. 906). Metacognitive sensations in the sense of Flavell may be conscious or ‘unconscious’ in the sense that it occurs automatically. This can be recognized on a behavioral level as: Copyright © 2015 by the author. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution International 4.0 License (CC-BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are properly cited. 118
Section 2.a. Presenter Articles
• • • •
‘Overflying’ (i.e. superficially scanning) Acting exclusively along schemas Looking for help Complete resignation
Effects of not responding to language with unknown symbols while being presented with mathematical formulas may be similar to that of ‘freezing behavior’ similar to other forms of reaction to stressful events and shock experiences. Freezing behavior is understood in biology and psychology as a form of reaction to trauma (Berceli, 2009), but the release of trauma or shock symptoms regarding symbol-shock will be another topic not explicitly addressed in this study.
A quasi-experimental setting In order to study the experience of an eventual symbol-shock, we presented young adults with three photos from a publication by a Swiss photographer (Pol, 2014): we asked them to describe the meaning of the symbolic elements and mathematical operators in the expression written in the right hand corner of the blackboard in the following picture.
Figure 1. Discussing physicists at CERN, (copyright Pol 2014). Method and Observations With background in Socratic psychology and more specifically in ‘Socratic consulting’ (Hake, 1992) our team looked for metacognitive strategies used by the young adults, mostly skilled craft trainees, by asking questions such as: ‘Describe what you see.’ and ‘What do these mathematical expressions on the black board mean to you?’ and finally ‘What do you think these expressions mean to the people in the pictures?’ The observed reactions of the trainees confronted with unfamiliar mathematical expression in this halfexperimental setting were quite different. We qualified quite a few different answers as possible forms of ‘shock-reactions’: • • • •
‘Seeing these kinds of formulas upsets me’; ‘Letters are for textbooks not for mathematics’; ‘Nobody can understand this stuff’; ‘May I leave the room?’ etc.
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Greber, R. (2015). Let's talk about math against symbol-shock.
Let’s Talk About Math Based on a Swiss reference textbook, ‘Mathematik im Gespräch’ (Discussing Mathematics, Liechti et. al. 1993), which navigates the teaching process of mathematics, we encouraged the trainees to read and paraphrase the mathematical expressions/operators one-on-one with a tutor for a basic comprehension. Furthermore we encouraged the trainees to use self-invented signs and symbols to comment on the mathematical expressions (Roam, 2008). To further the exposure to ‘symbol-shock’ our team encouraged participants to • • • • • • • •
Try to learn the alphabet of mathematical expressions on the internet (‘mathematische Notation’); Try to translate a mathematical expression into spoken natural language; Rewrite the mathematical term in blend of natural language together with mathematical symbols; Replace the most outstanding ‘shock-symbols’ with different and more intuitive symbols or words; Introduce symbols or pictograms with more intuitive meaning into the ‘translated’ formula Ask peers, teachers, parents etc. for help in understanding unfamiliar mathematical notations; Use intelligent computer software or smartphone and tablet apps; e.g. math-online on internet; Spend (more) time on mathematical tasks instead of avoiding and neglecting math-notations.
As a result of our explorative experiment our team was able to explain to the participants that science, especially mathematics and physics, may have much in common with a secret knowledge in occult science in antiquity. However today’s scientific knowledge and scientific know-how is accessible to everybody and is not limited to only scientists in ‘closed gate communities’. Then the team encouraged the participants to explore the many methods available for learning the basics of mathematics. These include illustrated textbooks, podcasts or open access internet-platforms such as Khan Academy (www.international.khanacademy.org). According to Frege (Begriffsschrift, 1879) sometimes it is helpful to regard mathematics and its notation as a form of logic which is illustrated in a well-made graphic novel by Doxiadis et al. LOGICOMIX (2009). Finally, as stated by Polya (2014), children and adults should simply be given the possibility to discover the meaning of mathematical problems by themselves and this should preferably occur in a form of a teacher-student dialog as learning is easier in joint efforts. Conclusion Symbol-shock – observed when people are faced with unfamiliar mathematical language and/or unknown mathematical symbols – may manifest itself in emotions such as fear, anger, and resignation etc. Symbol-shock is understood in this study as a combination or blend of cognitive and emotional states. To prevent the manifestation of symbol-shock we propose further exposure to mathematical formulas and symbols. In addition establishing a self-mental monitoring system to keep control of cognition and emotions can further benefit the young adults who lack the ability to comprehend the unfamiliar mathematical expressions and operators. By doing so, young adults may be able to give themselves the chance of not being overwhelmed by non-trivial mathematical notations. Therefore, let’s talk about math! References Berceli D., (2009). Evaluating the effects of stress reduction exercises employing mild tremors. A pilot study (Doctoral dissertation). Bohm E. (1996). Lehrbuch der Rorschach-Psychodiagnostik. Verlag Hans Huber: Bern. Doxiadis A. et al. (2009). LOGICOMIX, an epic search for truth. Bloomsbury Publishing UK. Adults Learning Mathematics – A Research Forum (ALM) 120
Section 2.a. Presenter Articles Flavell J. H. (1979). Metacognition and cognitive monitoring. A new area of cognitive-developmental inquiry. American Psychologist, 34(10), pp.906-11. Flavell J. H. (1985). Cognitive Development. New York: Prentice Hall. Frege G. (1879). Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought. Halle: Louis Nebert. in: Begriffsschrift Wikipedia. Hake, R. R. (1992). Socratic pedagogy in the introductory physics laboratory. Physics Teacher, 30(9), 546-52. Kahneman D. (2011). Thinking fast and slow. Allen Lane: Penguin. Kettler M. (1991). Der Symbolschock: eine Form metakognitiver Empfindung bei Lernenden im Prozess der Auseinandersetzung mit algebraischen Symbolen. (Doctoral dissertation). Kettler M. (1998). Der Symbolschock. ein zentrales Lernproblem im mathematisch-naturwissenschaftlichen Unterricht. Frankfurt/M, Europäische Hochschulschriften 11, Bd. 755. Liechti R. (1993). Mathematik im Gespräch. auf der Suche nach echtem Verständnis im Mathematikunterricht, Sabe AG: Zurich. Pol A. (2014). Menschen am CERN. Lars Müller Publisher: Zurich. Polya G. (2014). How to solve it: A new aspect of mathematical method. I. Roam D. (2009). The back of the napkin (expanded edition): Solving problems and selling ideas with pictures. Penguin: New York.
Resources Mathematische Hintergründe. Variable, Terme, Formeln und Identitäten. mathe-online. • http://www.mathe-online.at/mathint/var/i.html Mathematische Notation. • http://de.m.wikipedia.org/wiki/Mathematische_Notation#Mathematische Zeichen Term. In mathematical logic a term denotes a mathematical fact (recursively) constructed from constant symbols, variables and function symbols. • http://en.m.wikipedia.org/wiki/Term_(logic) Begriffsschrift • http://de.m.wikipedia.org/wiki/Begriffsschrift#Syntax and • http://gallica.bnf.fr/ark:/12148/bpt6k65658c
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