Catch the Mistake for Class Discourse and Productive Perseverance

We’ve all been there, a student brings you their attempt at a problem, and they ask you to find what they did wrong. You review it, can’t seem to locate the mistake, and instead pull out a clean sheet of paper and say “let’s start from the top and work it fresh.” Then you magically (in their eyes) arrive at the correct solution. Moral of the story: it’s harder to find the mistake than it is to work it correctly from scratch!

In the movie How to Lose a Guy in 10 Days, the advice columnist Andie (Kate Hudson) attempts to teach her readers about what NOT to do in a relationship. She illustrates with Ben (Matthew McConaughey) the common mistakes we sometimes make in relationships so that her readers can be aware of them and navigate relationships more successfully.

Counter-Example is an Effective Learning Aid

Don’t just show the correct approach, show a common mistake:

Correct (the sum of logs is the log of the product):

log_b(a) + log_b(c) = log_b(ac)

Example: log₂(8) + log₂(4) = log₂(32)

Beware of Common Mistake:

log_b(a) + log_b(c) ≠ log_b(a+c)

Counter Example:

log₂(8) + log₂(4) ≠ log₂(12)

Make it Simple and Assess Everyone—Not Just Some

Using examples in class with simple answers that students can hold up on their right hand if it’s positive, left hand if it’s negative and the absolute value is the number of fingers. For example, if the answer is 3, that’s a right hand three fingers, or if the answer is -4 that’s a left hand four fingers. All students are asked to hold up the answer and you can assess the entire class versus just assessing the one student called upon.

Find the exact value of (a) log₃81 (b) log₅ (1/25) (c) log₂1 

Answers (a) 4 (right hand four fingers) (b) -2 (left hand two fingers) (c) 0 (either closed fist)

Note on c: if a student holds up a 2 this is a common mistake. 

Class Discourse—Catch the Mistake(s)

Often when there could be more than one mistake, discourse is amplified.

In this example Steps 2 and 4 are both incorrect. . After discussion in which students share their thinking and ask clarifying questions, it is important to work (together) the same problem correctly so students can compare the steps when they are on their own. This process of metacognition (thinking through their thinking) helps to solidify a path forward.

Get Out of Class Problem—Helps Indentify Misconceptions

When a student is assigned homework, work a few problems and keep getting them incorrect, they often give up. One psychological approach is if you can do #20 you likely can successfully do #1–19.

If you can somehow carve out the last ten minutes of class try a get-out-of-class-early problem. Assign say #1–50 odd, then pick an even one in the middle to have students work and bring to you when done. If correct they get out of class early, or they can sit on their phone. If incorrect, they go back to their desk and try again. In the last two minutes work the problem and students will try to figure out the mistake they made and take notes to help them later. Here is an example-: #19 is assigned for homework so #20 is the get-out-of-class-early problem:

In my experience, after the first few weeks of this students who get it correct don’t actually leave early or get on their phone, they help others who have returned to their desks.

Challenge-Resist Starting with a Clean Sheet of Paper

The next time a student brings their incorrect solution and asks what mistake they made—if you can locate the misconception, it will help them remember the common mistake and the correct approach.