Caption: Sanjoy Mahajan demonstrates how not to calculate answers.
Want to know how new digital technologies and innovative approaches to learning are changing the face of education? One way is to follow the Office of Digital Learning's seminar series, xTalk. You can begin with this excerpt of an xTalk by Sanjoy Mahajan, MIT visiting associate professor of electrical engineering and computer science, reported by undergraduate Yuliya Klochan ’18:
If you've been to a high school math class recently, you probably remember seeing an inordinate number of calculators. You might’ve also observed students entering anything from 999*999 to sin(π/6) to (7-1) into the machines. The calculator habit is so strong that even 5+7 (=12) warrants machine help. And habits die hard.
To combat our calculator obsession, Sanjoy Mahajan suggests a radical departure from the calculator addiction: What if we knew the solution BEFORE the calculation? In Prof Mahajan’s words, “Never calculate without already knowing the answer!” Sounds bizarre, but let him explain, and you will be intrigued.
Sanjoy Mahajan sees a general trend of pragmatism in the United States, with students wishing to just solve the problem without asking the essential how or why. Science problems then become nothing more than a repeated and calculator-assisted application of a single formula. Unfortunately, even the most complex formulas can be wrong. Calculators can’t reveal that, but a better understanding, or insight, of the material can. In his May 19 xTALK, the Art of Insight, Prof Mahajan used several examples to illustrate the benefits to getting to the solution before reaching for a calculator.
EXAMPLE #1: consider being in a room with a paper towel freshly sprayed with perfume (as during Prof. Mahajan’s talk). The question: how long does it take for all audience members to smell the aroma?
Formal Model: Formally, the answer to that , is derived with a diffusion equation which, according to Prof. Mahajan, is a “big mess” that requires knowledge of calculus. Furthermore, the solved equation tells us that is seconds, or approximately ⅓ a year, while, in real life, the back rows smelled the perfume after 30 or so seconds! The official, calculator-enabled model for the problem is way off. Despite the complexity, it fails to provide a reasonable answer.
Insight: Add to the consideration a four-letter word: wind. With wind, or drift, you can start taking much bigger, though slower, steps to a solution instead of the small and fast steps of the diffusion model. This common-sense model has the benefit of search control: making search space much more tractable by expanding it.
See the full article for more examples.
Let’s recap now to consider all the benefits of reasoning, rather than calculating, the solution.
- Search control (making search space much more tractable by expanding it). Main example: perfume diffusion.
- Transfer (to other problems). Main example: perfume diffusion.
- Use of the (larger) visual cortex. Main example: solar flux.
- Robustness. Main example: solar flux.
- Low entropy/low cognitive load. Main example: well height.