Intuition in Science and Mathematics

What Hacker News says

1. rramadassFeb 2026
   Nice.

   There is also Intuition in Science and Mathematics: An Educational Approach by Efraim Fischbein - https://link.springer.com/book/10.1007/0-306-47237-6

   thread
2. rramadassSep 2025
   I am not sure what you are trying to argue with this gish gallop of a comment filled with strawmen. The usefulness of Geometry in visualizing and building intuition in Mathematics is a given. Equally, its limitations when it comes to higher dimensions is also known. But for a beginning student (the context of this thread) it is the former which must be emphasized before he has developed the requisite mathematical maturity to deal with the latter. You do not want to scare away students from the important practical subject of Linear Algebra by focusing too much on its complexity in the initial stages of learning.

   I highly recommend that you talk to/read the works of some mathematicians who are also involved with the research in the teaching of mathematics. There has been a lot of studies/research done on how to teach students build mathematical intuition using Geometry (and other graphical means) and in particular w.r.t. Linear Algebra. Some relevant references are given below for your edification; those by Harel are in particular, noteworthy.

   The farin and hansford book mentioned earlier is one of the best introductory texts for beginners since it starts with building geometrical intuition for the study of the linear algebra. Pair it with some more rigorous text like for example; Linear Algebra Pure & Applied by Edgar Goodaire and the student can comfortably learn a pure mathematical approach with a clear understanding of what he is doing. Goodaire is a "pure mathematician" who specifically starts with 2D/3D graphs and then transitions to n-space with all necessary definitions/theorems/proofs included in a very accessible writing style. This is the ideal approach to the teaching and learning of Linear Algebra i.e. use geometry (and other graphical means) in conjunction with algebra, and where applicable, include caveats/limitations, but always with the aim and focus of teaching concepts and not mere computation.

   References:

   1) Intuition in Science and Mathematics: An Educational Approach by Efraim Fischbein - https://link.springer.com/book/10.1007/0-306-47237-6

   2) The efficiency of visualization through geometry at mathematics education: a theoretical framework by Sefa Dundar et al. - pdf at https://www.sciencedirect.com/science/article/pii/S187704281...

   3) Should we teach linear algebra through geometry? by Ghislaine Gueudet-Chartier - pdf at https://www.sciencedirect.com/science/article/pii/S002437950...

   4) Geometric thinking in a n-space by Ghislaine Gueudet - pdf at https://shs.hal.science/hal-00529609/

   5) Students' understanding of proofs: a historical analysis and implications for the teaching of geometry and linear algebra by Guershon Harel - pdf at https://www.sciencedirect.com/science/article/pii/S002437959...

   6) Promoting Linear Algebraic Reasoning among Students: Affordances and Challenges by Guershon Harel - pdf at https://www.tandfonline.com/doi/full/10.1080/10511970.2024.2...

   Show morethread
3. 10renMay 2009
   He's saying design is trial-and-error, but it helps to have an approximately right model to start from. It seems to me that mathematics could lead us directly to the right model without mistakes... or is the truth that the same trial-and-error still occurs, but just in more abstract and general terms? Does that seem right to the mathematicians here?

   ... the nature of design, as they discovered, is so intensely cognitive, happening inside the mind at mind speed, the researchers could conceive of no useful tools to help in that process!

   This is exactly my experience. The closest is diagrams with pen-and-paper, but it doesn't help that much. The more difficult part is not solving the problem, but showing that you've solved it i.e. a proof, for all possible cases. Finding a proof of a solution is as much work (if not more) than finding the solution in the first place.

   It is by logic we prove, it is by intuition that we invent - Henri Poincaré

   The article is talking about consciously formulated conjectures, not about instantaneous insight, discussed here: p.62 of Intuition in science and mathematics: http://books.google.com/books?id=qqGWlEwWj5UC&pg=PA62...

   Show morethread