Inferentialism and the faulty logic of math wars - 2 views

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  #1 prattdc on 19 Jun 13

   

  Math wars have not been fought as fiercely here as in the US. Nevertheless, the debate about the relevance of algorithms has been a point of contest between 'back-to-basics' advocates, often influencing government policy, and 'progressives', sometimes ridiculed as the 'trendy left'.
  
  With the emphasis on 'fluency' in the latest drafts of the new maths curriculum, the debate is still relevant. Even so, my view is that the discussion has moved on - or at least should have moved on - and I read the article on the faulty logic of the math wars, as an attempt to move the debate on.
  
  I would argue that the difficulty with algorithms is not the algorithms themselves but the way that they tend to be taught without due care to enabling students to see the power and significance of the algorithms. Few students who simply attempt to learn the algorithms in rote fashion, as sometimes encouraged by the school system, will remember the procedure in the future. There is plenty of research evidence to show how students make errors and, when they do so, have no recourse to correcting their mistakes, allowing what should be self-evidently wrong answers to stand.
  
  I was intrigued that the article was written by philosophers; the debate is moving on under the influence of a philosophical stance called 'Inferentialism'. Indeed Brandom's Inferentialism is based on both Sellars and Wittgenstein, and developed through John McDowell.
  
  The Inferentialist account would regard the use of algorithms as an instance of a larger issue to do with the primacy of representations. The mathematics curriculum is populated by graphical and numerical representations to the extent that mathematics might appear to be no more than a list of such techniques with the result that competence in mathematics amounts to fluency in these techniques. According to the Inferentialist account of knowledge, the giving and taking of reasons is primitive, so that representations are invented in pursuit of those reasons.
  
  When the primacy of reasons is ignored, representations become meaningless. That is not to say that representations such as algorithms are not important - they are crucially important in mathematics - but they can only be understood in a web of reasons. Students need to be enculturated into the discipline of mathematics, coming to understand its power and its limitations, mapping out the territory that mathematics inhabits, so that algorithms and other representations are understood as having the power to get certain types of stuff done. I agree with the authors that there is something that we might call algorithmic thinking but I would associate it with an appreciation of the power of algorithms rather than remembering how to execute an algorithm.
  
  For our project, this debate is centrally important. The students who will be our 'clients' will have been failed many times in being expected to remember the algorithms and other representation to which they have been introduced without at the same time introducing them to the reasons for such things.

   

  <div class="cArrow"> </div><div class="cContentInner">Math wars have not been fought as fiercely here as in the US. Nevertheless, the debate about the relevance of algorithms has been a point of contest between 'back-to-basics' advocates, often influencing government policy, and 'progressives', sometimes ridiculed as the 'trendy left'. <br /> <br /> With the emphasis on 'fluency' in the latest drafts of the new maths curriculum, the debate is still relevant. Even so, my view is that the discussion has moved on - or at least should have moved on - and I read the article on the faulty logic of the math wars, as an attempt to move the debate on. <br /> <br /> I would argue that the difficulty with algorithms is not the algorithms themselves but the way that they tend to be taught without due care to enabling students to see the power and significance of the algorithms. Few students who simply attempt to learn the algorithms in rote fashion, as sometimes encouraged by the school system, will remember the procedure in the future. There is plenty of research evidence to show how students make errors and, when they do so, have no recourse to correcting their mistakes, allowing what should be self-evidently wrong answers to stand. <br /> <br /> I was intrigued that the article was written by philosophers; the debate is moving on under the influence of a philosophical stance called 'Inferentialism'. Indeed Brandom's Inferentialism is based on both Sellars and Wittgenstein, and developed through John McDowell. <br /> <br /> The Inferentialist account would regard the use of algorithms as an instance of a larger issue to do with the primacy of representations. The mathematics curriculum is populated by graphical and numerical representations to the extent that mathematics might appear to be no more than a list of such techniques with the result that competence in mathematics amounts to fluency in these techniques. According to the Inferentialist account of knowledge, the giving and taking of reasons is primitive, so that representations are invented in pursuit of those reasons. <br /> <br /> When the primacy of reasons is ignored, representations become meaningless. That is not to say that representations such as algorithms are not important - they are crucially important in mathematics - but they can only be understood in a web of reasons. Students need to be enculturated into the discipline of mathematics, coming to understand its power and its limitations, mapping out the territory that mathematics inhabits, so that algorithms and other representations are understood as having the power to get certain types of stuff done. I agree with the authors that there is something that we might call algorithmic thinking but I would associate it with an appreciation of the power of algorithms rather than remembering how to execute an algorithm. <br /> <br /> For our project, this debate is centrally important. The students who will be our 'clients' will have been failed many times in being expected to remember the algorithms and other representation to which they have been introduced without at the same time introducing them to the reasons for such things.</div>

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