In mathematics education our primary focus tends to be how to approach the concepts we teach, taking for granted the choice of subjects that enter the syllabus. Traditionally, the way the standards are determined follows a specific philosophical inclination: school mathematics is still predominantly galilean. It is viewed as a language with which to understand and describe objects and patterns we experience in the world. Math is still viewed as a vehicle for the expression of explanatory models of reality. That's how we define what knowledge is necessary (and thus mandatory): you can't describe movement without functions, you can't solve problems without algebra. Math is the list of things you (supposedly) can't go without in order to understand how the world works.

It can be argued that the developments in contemporary mathematics favor a more transformative, or "creative" approach to mathematical thinking. The mathematical foundations of visualization techniques, robotics, modern design, software engineering, cryptography, etc., do not deal with merely understanding, but mostly with producing new ideas, and new things. Math would be the way to express proposals. Still a language, but for writing "fiction".

My question is: assuming this change in perspective affects the way we teach math in school, how will it impact the way we define and organize the contents we will be working with, in the years to come? More specifically, how will we determine what is mandatory for the students of tomorrow?

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    $\begingroup$ We can only be certain that the math curriculum of the future will be ~50 years out-of-date. :-) $\endgroup$ May 29, 2015 at 14:13
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    $\begingroup$ I think the title question is interesting, depending on its interpretation; but the first two paragraphs in the post contain a lot of assumptions. I'm not at all sure what is meant by a more "creative approach to mathematical thinking." I find the italicized term very vague in this context; relevantly, see MESE 7661 and its links. $\endgroup$ May 29, 2015 at 21:59
  • $\begingroup$ I strongly disagree with the assertion that "It is viewed as a language..." We certainly don't teach mathematics as if it were a language--we teach it more as an innate "thing" and only explore the language aspects in higher level courses (upper division college courses). $\endgroup$
    – Jared
    May 31, 2015 at 9:53
  • $\begingroup$ @BenjaminDickman I admit that the question expresses a point of view that is far from consensual. I think it is suited for this format though, a bit of controversy can be tolerated if stated objectively. An answer could be used to refute my point of view, if that's the case, it would be an interesting discussion. As for the usage of the word "creative", it is not supposed to be a big word here. I suggest you take its meaning from context, not as terminology. I have edited the question to express that. $\endgroup$ May 31, 2015 at 12:35
  • $\begingroup$ @Jared You may be right about that, to some extent, considering what we do in elementary school. When I say that math is treated as language in school teaching (and I admit that this is a complicated issue) I do not mean that this is done explicitly, but implicitly. When you create an explanatory model for phenomena that come from experience (not from thinking alone), and express this model using mathematical rigor, you are taking mathematics as a descriptive language. This is a wonderful thing to do, don't get me wrong. I'm just saying that maybe this is not the only game in town, today. $\endgroup$ May 31, 2015 at 12:48

2 Answers 2


The (US) National Academy of Sciences wrote a book-length report, The Mathematical Sciences in 2025, available free.

Chapter 4 is "Important Trends in the Mathematical Sciences," whose table of contents I snapshot below.

  • $\begingroup$ While the National Academy Press is not charging for the free PDF version, from what I saw they do encourage people to create another internet account. Is there a way to get the report PDF without doing such a registration? Gerhard "Chooses Which Grids To Join" Paseman, 2015.05.29 $\endgroup$ May 29, 2015 at 16:31
  • $\begingroup$ @GerhardPaseman: I don't know. $\endgroup$ May 29, 2015 at 17:52

I think it is possible to teach innovation and change, or at least teach what it might look like. One could trace recent developments and hope to find a pattern that might allow a more direct path to a result, or one could take an approach similar to that of James Burke (Connections, The Pinball Effect) and try for unanticipated innovation by making many connections. In either case, a certain philosophical, historical, cultural, and mathematical foundation will be needed to appreciate or even just cope with either method. I recommend having a series of courses such that one path focuses on direct and anticipated innovation, and another path focuses on unanticipated innovation. You might find that you may not be able to predict the mathematics needed 50 years from now, but you can at least prepare for some of it. That preparation is what needs to be taught.

Gerhard "Might Look Back At This" Paseman, 2015.05.29


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