## Accessing Prior Knowledge in a Way That Uncovers Misconception - A Lesson

*If what we are doing in class today does not promote our understanding tomorrow, we have wasted our time.*

The order of operations and arithmetic with signed numbers combine to be the downfall of many high school math students. As with many things, remediation does not work. While exposing mistakes with these topics in applied contexts can be powerful, that also sometimes leaves students with the wrong impression that their new conceptual understanding is flawed.

For example, say a student is graphing a polynomial, factorable, function and they make sign errors. Their roots might be off and they’ll likely consider their new understanding to be flawed when in fact they’re just adding incorrectly.

With careful discussion and in an environment that encourages exploring mistakes, this is a fantastic way to shore up weaknesses that are prerequisite in nature. However, with a shy student or even a new student, this often just leads to frustration.

I’ve attached a link to a free PowerPoint activity/lesson that is designed to get students to explore combinations of arithmetic operations and the order of operations to arrive at an answer. It challenges their understanding of both integer operations and the order of operations in a way that does not just leave them wrong, but empowers them to change what they’re doing and make use of their wrong answers.

The activity uses the scary clown from the Saw movies. He wants to play a game. If you’ve never seen those movies they’re likely not your taste, but teens love them! The, *playing a game,* with this reference is powerful to them.

The game is that there are four-fours with spaces between, and they equal a number, like 1. Students can add in whatever operation signs and or parentheses they want in order to make the total equal 1.

An example is: 4 4 4 4 = 3

Students may add: + - × ÷ [ ]

So an attempt might be: -[4÷4]+4÷4

This is of course wrong, but there will be more gained from discussing why it is wrong than simply sharing what is right.

There are often many proper solutions, and if a student gets an answer quickly, have them see how many they can find. Another thing to do with a student that gets it too quickly is have them help another student, but not in a way where they share the answers but instead their thinking, helping the struggling student to arrive at their own answers.

Depending on the ability and enthusiasm of the class you can take one of the problems in the lesson and write four or five of that problem on the board and have people come up to the board and write in their arithmetic notation and parentheses. Then when all of the slots are filled, you can discuss what’s right and wrong, and why, as a class.

This would also make a fantastic white-board activity, although recording thoughts and realizations in their notes is very important, because:

*If what we are doing in class today does not promote our understanding tomorrow, we have wasted our time.*

When doing actual problems, the person needs to use parentheses frequently to avoid making mistakes. Also, calculators like parantheses in order to provide correct answers.