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Historical Word Problems - Practical or Abstract?


If we have learned anything from our mathematical past, it's that the world never splits cleanly into practical problems and abstract problems. A few simple examples of this might be the number π, which started very simply as a practical relationship between the circumference and diameter of a circle, but can also be calculated in a very abstract way with an infinite series (the Leibniz formula for π/4). Another example might be the many number theorist who prided themselves in pure abstraction of their field, only for a practical uses for their methods coming with the arrival of computers and the need for encryption techniques. 

This article brings up some interesting interplay between practicality and abstraction in ancient Babylonian word problems. Firstly it seems clear, from the examples provided at the beginning and throughout the text, that many word problems were inspired by what was familiar in their everyday lives, so in a sense they have a level of practicality in that they are related to a problem they might have faced in real life. However, they also move into the abstract, dealing with numbers much larger or simpler (integer answers) than would be realistic. 

To me, it seems the lines between 'pure' and 'applied', or between 'practical' and 'abstract' are constantly being blurred both in problems created to test and grow a student's understanding, and in real-world problems that need solving. In the created questions, an element of practicality is often added to an abstract problem to make it more tangible and relatable. In a real-world problem, the solution is often found by finding an abstraction of the practical problem (such as an algebraic formulation of the problem), solving that, and then applying the solution back to the original problem. 

There seems to be a constant interplay between pure and applied mathematics, and that idea should be inspiration in our modern context for a teacher to take lessons in the classroom out of the abstract (where lessons traditionally live) and into the practical, getting students to use their hands and their bodies and the world around them to engage in learning. 

Comments

  1. Amanda FritzlanNovember 10, 2022 at 4:17 PM

    Lovely articulation of the shifting relationships between pure and applied mathematics. I especially like your point that real world problems are often found through abstraction.

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