The Current New Math

Courtesy of Paul Kirschner (link and name corrected), by way of Greg Ashman:

17 Replies to “The Current New Math”

    1. Thanks, Terry. I think you may have linked that paper previously. All this stuff bores me, including the stuff with which I agree. I’m not convinced any of it is more necessary than a proof that rain is wet.

  1. Another reference from a different point of view.

    von Glasersfeld, E. (2013). Radical constructivism (1st ed.). Taylor and Francis.

    Many writers on constructivism do not define the term. Even when I have asked constructivists to define the term, they say “It’s complicated” … or something like that. I find this unsatisfactory.

    However, von Glasersfeld opens with a definition. As a reader, I know where he stands.

    “What is radical constructivism? It is an unconventional approach to the problems of knowledge and knowing. It starts from the assumption that knowledge, no matter how it be defined, is in the heads of persons, and that the thinking subject has no alternative but to construct what he or she knows on the basis of his or her own experience. What we make of experience constitutes the only world we consciously live in. It can be sorted into many kinds, such as things, self, others, and so on. But all kinds of experience are essentially subjective, and though I may find reasons to believe that my experience may not be unlike yours, I have no way of knowing that it is the same. The experience and interpretation of language are no exception.”

    1. What knowledge is in the heads of the person? How did it get there? Is Pythagoras’ Theorem in the head of a person?
      I haven’t read the paper, maybe I should. But perhaps what he says is all in my head and I just need to construct it for myself.
      How does von Glaserfeld argue against the EVIDENCE as presented, for example, by Kirschner? Does von Glaserfeld say how he learns?

      Different point of view …? Different planet, if you ask me. Planet Bullshit.

      1. @JF; Interesting that you pose this question about Pythagoras’s theorem. Attached are the slides and my notes from the recent MAV conference.


        1. Thanks, TM. I suppose there are some \displaystyle a \displaystyle priori assumptions the teacher must make:
          1) Every student is at least the equal of Euclid and is capable of such reasoning.
          2) There is sufficient time for every student to perform this reasoning.

          Maybe we should aim lower and assume every student is at least the equal of Gauss, who after all, constructed the sum of an arithmetic sequence at the age of 8.

          Under these assumptions, who needs a teacher?

          Is there a distinction between constructing knowledge and constructing understanding based on knowledge?

          1. My paper is not about teaching – it’s about how we should portray mathematics in society which was one of the themes of the conference.

            However, in answer to your first question, I see teachers as part of the external world who play a key role in the process of gaining knowledge and understanding. The objects in the external world that enable students to develop understanding of say Pythagoras’s theorem are more than simply the triangular shape on the house in the slide; they include the teacher, Euclid’s Elements, fellow students, the Internet …

            These all help the student to develop his or her own understanding through reasoning.

            Thanks for reading the paper and your thoughtful comments.

            Best wishes for 2022.

    2. Very interesting. I recently had an email exchange with Tony Gardiner, where he was suggesting that Freudenthal needed to be distinguished from modern “constructivists”. According to Gardiner,

      [Freudenthal] realised that we ALL have to “construct” the “universe of mathematics”
      in our heads. But his was NOT a subjective universe.
      He devoted his efforts to
      – helping ordinary kids construct
      – the one true universe
      – in their own heads.

      I replied, suggesting that modern “constructivists” didn’t doubt the objective mathematical universe, just the manner in which children could best learn to see it and understand it. Tony dismissed my suggestion and, of course, he was correct, as evidenced by the nonsense to which Terry has just referred.

  2. This blog post perfectly concluded the last page of 2021.
    It explains the crux and main issues posed to our modern education.
    The constructivism approaches (inquiry approaches, problem-solving based approaches, etc.) are not wrong themselves, but it may be dangerous if they are overused and/or misused in our contemporary classroom settings, especially at early stage of schooling where students are expected, in their best interest of undertaking future education/strengthening employability, to establish robust and correct understanding to basic concepts, skills and proficiencies under well-structured guidance and supervision. Indeed, these are like lobsters and foie-gras but here I quote from an expert teacher with extensive knowledge and experiences: “meat and veggies keep you healthy”.

    Thank you everyone for your continuous passion, discussions and shared insights, and I wish you all – Happy New Year 2022 with all best wishes!

    1. Indeed. Water is important for life. But too much water and you die.
      The right tool for the right job. And there are so many tools …

      Ditto with the best wishes to all for 2022.

    2. Huh. All I wanted was for people to watch the funny tweet. But if people want to engage in a round of constructivist bashing, that’s fine with me.

  3. The views of von Glasersfeld above have some merit. My view of the world has been constructed by me – through reading, thinking, being taught etc. And my view has much in common with the views of many other people. Nevertheless, my knowledge is subjective knowledge. As teachers, we should be aware of this. We often say that students who do not agree with our views have misconceptions. But the debate about constructivism raises the question as to whether there is objective knowledge at all, and it’s not a trivial question; Popper wrote about it.

    BTW, one of my Year 9 students asked this week “But how do we know that mathematics is true if it has been invented by human beings?”

    1. Terry, if you want to debate whether or not there is objective knowledge, go ahead. If you think that question can shed any light on the current problems with maths education, you’re nuts.

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