Freudenthal (1981)

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Major Problems of Mathematics Education


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In this article, Hans Freudenthal articulates 11 major problems of mathematics education. (Most of Freudenthal's spellings are preserved below.)

  1. Why can Jennifer not do arithmetic?
    In this problem, Freudenthal critiques educators from trying to diagnose students' difficulties as a doctor would diagnose an illness, claiming educators too often diagnose for what is wrong, not why something is wrong. Instead, Freudenthal suggests we rely on the "vast resources of human experience" (p. 135) to build "paradigmatic cases, paradigms of diagnosis and prescription, for the benefit of practitioners and as bricks for theory builders" (p. 135).
  2. How should children learn?
    Freudenthal extends this to "How should people learn?" and how we learn to observe learning processes. He suggests answers lie in "observing learning processes, analysing them and reporting paradigms — learning processes within the total educational system, learning processes of pupils, groups, classes, teachers, school teams, councillors, teacher students, teacher trainers, and of the observer himself" (pp. 136-137).
  3. How to use progressive schematisation and formalisation in teaching any mathematical subject whatever?
    This question receives Freudenthal's longest reply, including illustrations of what he means by schematisation. Freudenthal claims that the "history of mathematics has been a learning process of progressive schematising, (p. 140) but claims that "Youngsters need not repeat the history of mankind but they should not be expected either to start at the very point where they preceding generation stopped" (p. 140). He clarifies that schematising should be seen as a psychological progression or network of possible progressions.
  4. How to keep open the sources of insight during the training process, and how to stimulate retention of insight, in particular in the process of schematising?
    This problem tackles the problem of learning math conceptually versus procedurally, which Freudenthal contrasts with the terms "insight" versus "training." He feels that more mathematics has been learned by insight than we typically recognize, but it becomes a problem when the insights of the learner is superseded by those of the teacher and textbook author. Freudenthal feared that too much training, particularly premature training, endangers the ability for the learner to retain their first insights and actively use them when schematising.
  5. How to stimulate reflecting on one's own physical, mental and mathematical activities?
    Freudenthal believed that reflection could help learners retain their insights, but purposeful reflection is lost in the learners' efforts to master rules that seem disconnected to those insights.
  6. How to develop a mathematical attitude?
    Freudenthal claims attitude can be described by example, but we lacked the language to describe problems, strategies for changing perspectives, understanding precision, knowing when mathematics might not be applicable, and "dealing with one's own mathematical activity as a subject matter in order to reach a higher level" (p. 143). Freudenthal uses these examples to warn against what he says is the usual mistake: "testing a mathematical attitude by asking questions about an attitude towards mathematics (p. 143).
  7. How is mathematical learning structured according to levels and can this structure be used in attempts at differentiation?
    In this problem Freudenthal addresses the fact that learners learn differently and at different rates, yet need to work together.
  8. How to create suitable contexts in order to teach mathematising?
    Freudenthal focuses this problem more squarely on mathematical subject matter and the need for contexts which to mathematize. He briefly describes one of the foundations of Realistic Mathematics Education, that the "'real world' is represented by a meaningful context involving a mathematical problem" (p. 144) with 'meaningful' meaning a context that's meaningful to the learners.
  9. Can you teach geometry by having the learner reflect on his spatial intuitions?
    Freudenthal asks that learners become more aware of their intuitive grasp of space as a way of understanding geometry, but questions how this leads to formal subject matter knowledge.
  10. How can calculators and computers be used to arouse and increase mathematical understanding?
    This problem is less about using technology to calculate, but rather how technology can be a tool that helps learners explore mathematical relationships and fundamental concepts.
  11. How to design educational development as a strategy for change?
    Freudenthal claims that educational problems need to be solved in the educational process, not in the laboratory, and that learning will be a slow and social process. Curriculum development, says Freudenthal, is not enough. Our problems need to be addressed longitudinally, across levels, and in context.

Freudenthal continues the article by commenting on the influence of textbooks and teacher training. Freudenthal saw teachers relying on textbooks because they've been inadequately trained, but believed good textbooks — ones that mathematized the environment and led both teachers and learners along paths of retention and insight — were worth developing. Teacher training, on the other hand, was inadequate because while it exposes student teachers to short term learning processes, it's not possible to give preservice teachers deep understanding of long term learning processes.

Lastly, Freudenthal comments on educational research, much of which he did not find useful. However, he believed the quality was improving, even if the sheer quantity of research made the quality difficult to find. He hoped for less research of education, and more research in education.


Freudenthal, H. (1981). Major problems of mathematics education. Educational Studies in Mathematics, 12(2), 133–150. doi:10.1007/BF00305618
author = {Freudenthal, Hans},
doi = {10.1007/BF00305618},
journal = {Educational Studies in Mathematics},
number = {2},
pages = {133--150},
title = {{Major problems of mathematics education}},
url = {\%2FBF00305618},
volume = {12},
year = {1981}