Mathematical Truth in the Digital Age

When I saw the title “Ideas in the Ascendant” of an article in the Chronicle of Higher Education, followed by the synopsis — “In an online age, truth is more unbundled than ever. That makes higher education more important than ever” — I did not expect it to have been written by a mathematician.  The author, John Swallow (whose textbook, Exploratory Galois Theory,  I’ve used once and may again), contrasts his younger collaborator Adam’s research style with his own.  “Adam’s intellectual delivery system was need-to-know, just-in-time.”  Swallow remarks that “ideas à la carte are less dangerous in certain disciplines, like mathematics.”  I recommend reading the whole article, and not just for the clever metaphors.  But I’m not as sanguine as the author about, for example, the influence of calculators.  This may be because he’s focusing on those few students who will carry mathematical knowledge forward in the future, while I’m still worrying about the ones who think sine and cosine are just buttons to push.  I do think he’s on the mark about our students’ approaches to knowledge, and about our responsibility to figure out how to help them find the truth in what they discover.


About Priscilla Bremser

Professor of Mathematics Middlebury College
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2 Responses to Mathematical Truth in the Digital Age

  1. I enjoyed the article — this part really spoke to me: “I showed Adam a graduate textbook that I thought might be helpful to our project. What did he do? Just as he might click from Web page to Web page, he flipped through the pages, looking for theorems. If a potentially relevant one used terms or concepts he did not know, he learned about them.”

    Now that I’ve become so used to the instant searchability (I seem to have invented that word) of the internet and electronic documents, trying to find anything in a traditional paper book is frustrating. When I was writing my thesis, I did a lot of cherry-picking from textbooks and then running straight to the internet for further explanation — I learned how to do continued fractions from a combination of Prof. Schumer’s book and Wikipedia. I always had the “received authority” of the textbooks on hand to make sure I wasn’t going horribly wrong, but for many things the internet was just more efficient.

    I still think that knowing how to do research from print sources is an important skill. I assigned a research paper for one of my high school classes this year, and I required them to use at least one print source, but in retrospect I’m questioning that decision — it was a social justice class, and several of them chose topics that were so new they had difficulty finding anything relevant in our town’s slightly outdated library. Now I think they might have been better served to spend more time learning to use online databases like JSTOR rather than futilely searching for a print source of limited usefulness. I realize I’m straying from the math, but for me it’s all intertwined — my school takes a very traditional view of most disciplines, and integrating technology continues to be a huge struggle, since some teachers and administrators have all the fears discussed in the article. I’m torn between the different points of view.

  2. Rob Root says:

    Thanks, Priscilla, for sharing this thought-provoking piece. There is a lot here to think about. I have to say that Swallow is more sanguine about several important issues than I am. His blithe dismissal of the impact of calculators on math instructions as a problem solved does not match with my own experience, by that is small potatoes in comparison with his comfort with a la carte treatment of ideas.
    Mathematicians need to be particularly aware of consequences of assumptions, of course. And we live in a society where assumptions are made with frighteningly little reflection. By “reflection” I mean rational consideration, a habit of mind that schools have the potential to cultivate. Swallow says “for students to manage ideas successfully, they need guidance in developing their capacities for argument and judgment. Such guidance is neither easily given nor easily received, but it is essential.” I believe that the rarity of the guidance is understated even by Swallow’s pessimistic assessment, particularly if we discuss specifically numerate argument and judgment.
    My opinion is colored by my children’s experience in the local public high school, where even AP literature classes are being taught in classes of 25-30. If it is possible for a teacher to inculcate the desired kind of judgment in a class that large, I’ve not seen it. And there is no attempt to incorporate even elementary numeracy into writing and discussion, so far as I can tell.

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