Tina Cardone, Geometry Lead, & Gabriel Rosenberg, Curriculum Writer
There is no doubt that proof plays a central role in the human endeavor of mathematics, but there remains much debate on what role it should play in high school mathematics. At least two standards for mathematical practice in the common core focus on this concept. Certainly MP3, “Construct viable arguments and critique the reasoning of others”, is about the need for students to be able to write their own proofs and to analyze the proofs of others. MP6, “attend to precision” goes deeper, though, by noting the need for precision, including the use of clear definitions, when communicating their reasoning. This is what we mean by rigor in mathematical proof.
In a previous post we described the stages of the proof cycle which students engage in throughout our geometry curriculum. The cycle includes: experiment, conjecture, convince themselves, convince others, read proofs, critique proofs, and write proofs. One of the most challenging steps of this process is supporting students to develop the precision of language required to write a rigorous proof. One way we help students achieve rigor is by having students approach writing detailed proofs much like they write essays in their other classes. They write an outline, write a rough draft, and with the help of peers revise their proofs to attend to precision in a final draft.
One of our favorite examples of this revision process occurs toward the end of the unit on congruence. Students choose which of a few conjectures about quadrilaterals they would like to prove. For example, they might choose “If the diagonals of a quadrilateral both bisect each other and they are perpendicular, then the quadrilateral is a rhombus.”
- Outline: At this point students are encouraged to think about what information they have and what it is they need to show. They interpret the conjecture and add details that will help them to write a proof.
Students may realize that it is helpful to give names to the vertices of their general quadrilateral. Perhaps quadrilateral MNOR has diagonal MO which is the perpendicular bisector of NR and NR is the perpendicular bisector of MO. Then they can think about what it means to be a perpendicular bisector. In doing so they are taking the opportunity to make sense of problems and persevere in solving them (MP1).
- Rough draft: If students and teachers wish to use the two-column structure this is the place for it. The final drafts of proofs are all written in narrative form in this curriculum because this form matches the discussion students might have to convince their partner. They also match the way mathematicians write proofs. While students may use other formats to support their organization, it is important that students can see the flow of reasoning that exists in a well-written proof.
A student’s rough draft might involve the idea of reflecting across MO to show that side NO is congruent to side RO and side MN is congruent to side RN since reflections are rigid motions. In the same manner, reflecting across NR would show the other pairs of adjacent sides are congruent.
- Peer revision: At this point students exchange drafts of their proofs and see whether they convince a partner. Here students are both constructing viable arguments and critiquing the reasoning of others (MP3). In previous lessons students learned to be skeptics that push their partners to be more precise in their proofs (MP6).
For example, a partner reading the draft above might ask, “How do we know that reflection across MO takes side NO to side RO?” A more precise rigorous proof would note that since point O is on the line of reflection it remains fixed and since MO is the perpendicular bisector of NR the image of point N must be the point R.
- Final drafts: Once both partners are convinced, they work together to write up final drafts of their proofs. These are again in narrative form using sentences.
As we noted in the previous blog post about proofs, this is just one part of the process and not every proof needs to be written rigorously. Teachers often wonder what level of precision and rigor they should require from their students. While the sample responses in the IM curriculum demonstrate a high degree of precision and rigor, our expectation for students is that they receive meaningful feedback and revise their proofs throughout the unit. While it’s important to attend to precision, we don’t want students to get lost in the details. Instead, we focus on students asking “is this clear and convincing?” and revising their proofs to meet that standard.
When do you invite students to improve the rigor of their statements?
How do you build time for drafting, getting feedback, and revising into the proof process?