A Common Source of Student Mistakes

Many students who pattern-match will tend to prefer solutions requiring fewer and simpler operations, especially if those solutions yield ballpark-reasonable results.

Cross-posted from here.

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General Framing

Many students who pattern-match (as opposed to reasoning from first principles) will tend to prefer solutions requiring fewer and simpler operations, especially if those solutions are

• "directionally correct" in the sense that they make a quantity increase or decrease as expected, and
• "order-of-magnitude correct" in the sense that the final result is on a sensible order of magnitude compared to expectations.

It’s essentially the result of applying Occam’s razor in the case when common sense has granted you some significant but imperfect level of predictive ability. (This might also be characterized as the principle of least effort.)

Examples of Predicted Mistakes

Here are some examples of mistakes that would be predicted by this framing in the context of percents.

If the price is $\$500$ after a $20\%$ discount, then what was the original price?

Well, the original price was obviously higher, and the simplest way to get a sensibly higher number using the quantities given in the problem is to just add $20\%$ of $\$500$ to the original price.

If a $\$500$ price is discounted by $10\%$ today and again by $20\%$ tomorrow, what is the price tomorrow?

Well, the discounted price is obviously lower, and a simple way to get a sensibly lower number using the quantities given in the problem is just to total up the discounts ($10\% + 20\% = 30\%$) and then apply that discount.

A Less Common Type of Mistake

When applying a discount, you might also see a student divide by the markup multiplier. For instance, if a $\$500$ price is discounted by $20\%,$ you might see a student divide $\$500 \div 1.2.$

I used to see this occasionally with students who were first learning about percents. If a student gets used to solving two-step problems about percent increase (write $1.\textrm{something}$ and then multiply or divide), then it’s easy for them to mistakenly apply the same technique to get a directionally-correct, order-of-magnitude correct answer to a percent decrease problem. That said, this mistake tends to get hammered out quicker than the previous ones because the corresponding problems are more basic / foundational and therefore students get way more practice with them.

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