Mathematicians have been figuring out the mathematical truth for millennia. Can this possibly be relevant for today’s students? Yes!
Mathematical argumentation can make your classroom more joyful and engaging. Going beyond just rules to memorize, students are given the opportunity to make sense of those rules and to convince others of their ideas. Students get to play with mathematical ideas and take ownership of them in a way that often delights them. You’ll most likely feel a boost yourself. One teacher we worked with proclaimed that every Friday was argumentation day, and her class eagerly looked forward to it.
To engage students through argumentation, follow these steps:
- Choose a topic on which there are likely to be differing ideas. For example, most students know that if you square a whole number greater than 1, the result is a larger number. But what happens if you square a fraction? Will the result be larger, smaller, or does it depend? The answer is not immediately obvious to most sixth graders and that makes it a good candidate for argumentation at that grade level. For other topics, you can use your state standards to come up with likely candidates for the grade you teach. Or look for places where a textbook may state some rules but there isn’t much explanation of why they work—that’s what you can have students figure out.
- Ask students to play around with examples and make a conjecture— their best guess at what is true. For example, students may try easy fractions such as 1/3, and find that (1/3)2 is 1/9. Since a ninth of a whole is smaller than a third of the same whole, 1/9 is less than 1/3. Students may conjecture based on this that whenever you square a fraction, the result is less than the fraction you started with. Now you have something students can argue about.
- Start by finding out which students agree or disagree by a show of hands. We don’t find out the mathematical truth by voting, but you will get a sense of whom to call on to add to the argument. Ask students who agree that the conjecture is true to go beyond the one example: How do we know it is always true, no matter what fraction we choose?
- Recommend representations. If students aren’t sure how to go beyond giving a couple of examples, which can never be fully convincing of what’s always true, you can suggest that they try to think with other representations. Give them some individual or paired think time with the representation of their choice. Can they make a table and look for patterns? Can they draw a diagram? Make a graph? Use algebra? Try not to tell them exactly how to use a representation—let them try their own ways, which may be ways you have never thought of. Have students compare different representations they used—connecting representations is a great way to increase conceptual understanding and you’ll increase engagement as students discuss their own representations.
- Keep the argument going with hand signals. Have students use a fist-over-fist motion to indicate they want to build on another student’s ideas, not just start a new argument. Keep asking for agreement and disagreement by a show of hands. Students can raise 1 to 5 fingers to indicate how convinced they are.
- Be sure to call on those who disagreed with the conjecture. That’s where great counterexamples come from—students thinking of particular fractions that break the general rule that most of the class thinks is true.
- Have students summarize the argument and the conclusion. For example: We decided that when a fraction is less than 1, if you square it, the result is less than the fraction. This is because if you take a fraction a/b, where a is less than b, and square it, you get aa/b•b. The denominator is is getting even larger than the numerator, since you are multiplying a greater number, b, by itself. But not all fractions are less than one! If you square a fraction greater than 1, such as 3/2, the result is larger, by the same argument turned “upside own”. There’s one other case: when a fraction is equal to one. When you square it, the result is still one.
Follow these steps for a topic appropriate for the grade level you teach, and support students in finding out the mathematical truth for themselves. You may find that their truth seeking expands to outside the classroom.