One goal in every effective math classroom is to get students to talk about mathematics. We want our students to analyze, question, and critique other students thinking. One way to encourage this type of discussion to take place is to force students to choose sides.
This can be accomplished by providing questions that may result in students' getting different answers based upon how they solved the problem or by asking students to examine and analyze common misconceptions in mathematics. The examples below show both.
Defending your answer and critiquing the reasoning of others can be accomplished by giving a problem that often produces two or more different examples. Ask students to solve the following problem individually:
A man named Sam bought a horse for $50. He then turned around and sold it for $60 to his brother Jake. Then he bought the horse back from Jake for $70. A week later he sold the horse again to his cousin Bob for $80.
What is the financial outcome of these transactions? Did he make or lose money? How much? (Ignore cost of feed for the horse, cost of boarding, etc.)
As students solve the problem, the teacher roams and looks for all the different answers. The most common answers tend to be +20, +10, break even, -10. Once all students are finished ask them to move to the corner of the room that goes with the answer they calculated. Once the students are grouped the goal is for them to discuss the answer they calculated and formulate a way to convince the other groups that their answer is correct. Groups then share out and students are free to move from one group to another if they are convinced. After all groups have shared, act out the problem using play money to determine the correct answer.
Looking at common misconceptions in math is another way to get students to discuss and learn valuable concepts. Below are a few examples:
Are these both ABB patterns? If "yes", go to this side of the room, if "no" go to the other side. Plan with your group how to convince the others.
Do the following represent ¾
It is amazing on a linear model students don't tend to naturally grasp that ¾ is less than one; therefore it should be ¾ of the way between 0 and 1. I have also noticed students tend to model ¾ this way:
Even in a set model, many students are not understanding it is three out of four, not three out of a total of seven items. Hmm....
Finding student misconceptions are easy. Whenever you are introducing a new concept and your are finding out what they know, roam and look for opportunity to discuss and make students choose and then justify their belief. Explaining and critiquing the reasoning of others not only will improve the mathematical understanding in your classroom but will also help students prepare for the Common Core Standards in Math.
If you have other ideas of how to improve students' reasoning, I would love to hear your ideas.
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