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The MADIF papers 2000-2016 [email protected] Content Introduction ........................................................................................................................................... i Killer-Equations, Job Threats and Syntax Errors................................................................................. 1 Student-mathematics versus teacher-Metamatics .............................................................................. 19 Mathematism and the Irrelevance of the Research Industry .............................................................. 31 The 12 Math-Blunders of Killer-Mathematics ................................................................................... 41 Mathematics: Grounded Enlightenment - or Pastoral Salvation ........................................................ 49 Discourse Protection in Mathematics Education ............................................................................... 57 Post-Constructivism ........................................................................................................................... 65 Golden Learning Opportunities in Preschool..................................................................................... 73 Calculators and IconCounting and CupWriting in PreSchool and in Special Needs Education ....... 81 Grounding Conflicting Theories ........................................................................................................ 89 Introduction Swedish school mathematics always fascinated me. Each second year Sweden arrange a Biennale where mathematics teachers from kindergarten to college can meet to share knowledge through exhibitions and inform themselves about new trends and ideas, and listen to foreign or local researchers having met the day before at the MADIF conference, the Swedish Mathematics Education Research Seminar arranged by the Swedish Society for Research in Mathematics Education. Furthermore, In 1999 the Swedish government decided to establish and gracefully fund a national resource centre for mathematics education, NCM, describing its task to ‘co-ordinate, support, develop and implement the contributions which promote Swedish mathematics education from preschool to university college’. What a bright future for Swedish mathematics, I thought and decided to contribute with a paper at each MADIF conference and a general talk or an exhibition at each biennale. My MADIF2 paper introduced postmodern counter research looking for hidden possible explanations for the problems in mathematics education within mathematics itself and warns against ‘killer-Equations’ and syntax errors. Furthermore, the paper suggests an alternative mathematics curriculum for the new millennium replacing the traditional Top-Down approach with a more user-friendly Bottom-Up approach. The paper was accepted for a full presentation. However, I soon realized that it was almost impossible to establish e dialogue with the NCM and with Swedish researchers, so at the MADIF4 conference I presented a paper called ‘Mathematism and the Irrelevance of the Research Industry’ warning against supporting the irrelevance paradox in mathematics education research described by the following observation: ‘the output of mathematics education research increases together with the problems it studies - indicating that the research in mathematics education is irrelevant to mathematics education’. The paper demonstrates how to avoid mixing up mathematics with mathematism, true in the library but seldom in the laboratory. Although accepted for a full presentation, nothing happened afterwards, so in my MADIF5 paper I decided to be much more specific by warning against twelve blunders of mathematics education. The reaction to this paper was to reduce the presentation to a short communication. In my MADIF6 paper I draw attention to the difference between North American enlightenment schools wanting as many as possible to learn as much as possible, and European counter- Enlightenment Bildung schools only wanting the elite to be educated. In the enlightenment school enlightenment mathematics is grounded from below as a natural science enlightening the physical fact many. In the Bildung schools pastoral ‘metamatism’ descends from above as examples of metaphysical mystifying concepts. The paper was rejected based upon a review process that allowed decisions to be made without specific reference to the paper reviewed. So in my MADIF7 paper I warned against what I called ‘Discourse Protection in Mathematics Education’ and against reducing a constructive review process to what I called ‘Moo Review’ and ‘Tabloid Review’ using only one word or one sentence. Again the paper was rejected. One would expect the massive Swedish investment would show in the PISA scores. Here Sweden scored 502, 494, and 478 in the 2006, 2009 and 2012. Three consecutive numbers allow calculating the yearly change and the change to the change, which in the case of Sweden is -1.3 in 2006 changing yearly by -0.9 bringing the Swedish score to the zero level in 2038 if not changed. At the same time research had demonstrated the positive effect of an early start in mathematics, so to be helpful to the Swedish research community I wrote a paper describing the golden learning opportunities in preschool accompanied by a YouTube video’ Preschoolers learn Linearity & Integration by Icon-Counting & NextTo-Addition’ (https://www.youtube.com/watch?v= R2PQJG3WSQY). The paper presents mathematics as natural science about the natural fact Many. To deal with Many we count and add. The school counts in tens, but preschool also allows counting in icons. Once counted, totals can be added. To add on-top the units are made the same through recounting, also called proportionality. To add next-to means adding areas also called integration. So accepting icon-counting and adding next-to offers golden learning opportunities in preschool that are lost when ordinary school begins. And again, again the paper was rejected, this time however without using moo- or tabloid-review. In the PISA report Denmark scored 513, 503 and 500 giving an initial yearly change of -4.5 in 2006 changing yearly by 0.8 bringing the Danish score to 629 in 2030 if not changed. However, Denmark has not significantly increased its research activity. So the Danish success and the Swedish melt-down both indicate the correctness of the irrelevance paradox: More research creates more problems. Consequently I suggested a two year no-research pause in Sweden. It was declined because researchers had found a new research paradigm, Design Research, they hoped would change the situation in a positive way. Design Research bases its designs on existing theory. However, in conference presentations, disagreements between conflicting theories were simply ignored or denied. And not differentiating between grounded and ungrounded theory will hardly prevent the Swedish melt-down. So, to once more offer my assistance, instead of writing yet another paper that will be rejected yet again because of discourse protection, I have decided that my contribution to the MADIF 10 conference in 2016 should be a YouTube video similar to the Paul and Allan debate on postmodern mathematics education (https://www.youtube.com/watch?v=ArKY2y_ve_U), inspired by the Chomsky-Foucault debate on human nature (www.youtube.com/watch?v=3wfNl2L0Gf8), this time called ‘Grounding Conflicting Theories to avoid the Irrelevance Paradox creating the Nordic Math Melt-Down - an invitation to a dialogue on Mathematics Education and its Research’. One prominent person within the research community has declined to take part in the dialogue, but hopefully other persons will accept their responsibility and be willing to enter into a fruitful dialogue to prevent the Swedish melt-down to become reality. Money does not solve the problem, dialogue between conflicting theories does. The MADIF papers For the MADIF 2 conference in 2000 I wrote the paper ‘Killer-Equations, Job Threats and Syntax Errors, a Postmodern Search for Hidden Contingency in Mathematics.’ The abstract says that modern mathematics is facing an exodus problem: an increasing number of students are turning away from mathematics in school, and from math-based educations within science and engineering after school. Modern research looks for explanations within human factors: students, teachers and cultures. Postmodern counter research looks for hidden possible explanations elsewhere, in this case within mathematics itself. This study identifies unnoticed syntax errors within mathematics and a problematic Top-Down practice of allowing killer-equations into the classroom. Also the study reports on a successful changing of this practice and reflects upon why a Bottom-Up approach might be more user-friendly than a Top-Down approach. The paper contains chapters called: The Difference between Modern Research and Postmodern Counter Research, Killer-Equations in Paradise, Designing an Alternative: Rephrasing Equations, Practising the Alternative, Evaluating the Alternative, Why Might Bottom-Up Mathematics be More User-friendly?, Why Might Bottom-Up Mathematics be Unrecognised? - Rephrasing Mathematics, Mixing Different Abstraction Levels Creates Syntax Errors , Abstraction Errors, Equations Can Also be Solved by Reverse Calculations, Bottom-Up Mathematics Education Through the Social Practices that Created Mathematics, The Social Practice of Bundling and ii Stacking, The Social Practices of Measuring Earth and Uniting

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