Derive and explain geometric arguments and proofs.
SLO 1 Summary
Proof is a cornerstone of mathematics, and a GeT course should enhance a student’s ability to read and write proofs of theorems, apply them, and explain them to others. Geometry offers one of the best opportunities for students to take ownership of proof in both the undergraduate and secondary curriculum. Because geometric proofs come with natural visualizations, they provide a rich environment for students to think deeply about the arguments that are being made and to make sense of each statement. Students should understand that proof is the means by which we demonstrate deductively whether a statement is true or false, and that geometric arguments may take many forms. They should understand that in some cases one type of proof may provide a more accessible or understandable argument than another.
SLO 1 Narrative
Proof is at the heart of mathematics, and geometry is a place that offers robust opportunities for students to take ownership of proving in both the undergraduate and secondary curriculum. Geometric proofs are the result of investigation and conjecture, often stemming from natural visualizations; therefore, they offer the opportunity to think deeply about the arguments that are being made and try to understand what is being communicated. Prospective geometry teachers need to be able to understand different types of proofs, such as synthetic (from axioms), analytic (using coordinates), and proofs using transformations or symmetries. They should also be able to communicate proofs in different ways (two-column, paragraph, or a sequence of transformations).
Proof is a central idea in many state K-12 mathematics standards. While not every state has adopted or uses the Common Core State Standards for Mathematics (CCSSM), many have written new mathematics standards that are very similar to these standards. In the CCSSM, the first mention of proof is in the eighth grade geometry standards. These standards focus on using rotations, translations, and reflections to demonstrate that two figures are congruent to one another, and students are expected to be able to explain a proof of the Triangle Angle Sum, as well as the Pythagorean Theorem and its converse. In the CCSSM high school geometry standards, students are expected to “understand congruence in terms of rigid motion” and “prove geometric theorems” (National Governors Association for Best Practices[NGA] & Council of Chief State School Officers [CCSSO], 2010).
For secondary mathematics teachers to be prepared to teach proof, they need to have numerous and varied experiences to develop a deep understanding of proof. Mathematics teachers need to experience geometry proofs from the student’s perspective so they can empathize when their own students struggle. It is also essential that they can choose the most accessible type of proof for the situation.
The geometer David Henderson (2006) argued that we should be teaching “alive mathematical reasoning,” in which we view proofs as “convincing communications that answer—Why?” (p. 13). Under this view, the role of a proof is not merely to show that something is true but to clearly communicate to others why it is true. Most mathematicians would agree that explaining why something is true is an important value in mathematics, but it is not a perspective that we always share with our students when teaching them about proof. There are sometimes tradeoffs involved in choosing how to present material between making it clear why something is true and using easily adaptable proof methods. It is valuable to show GeT students a range of possibilities and ask them to evaluate the pros and cons as a class.
The reasoning used in arguments can be deductive or inductive. An argument that uses deductive reasoning, a deductive argument, is one that proceeds from a given set of assumptions to the logical consequences of those assumptions, while an argument that uses inductive reasoning, an inductive argument, is one where a pattern is noticed and assumed to continue. In mathematics, we primarily value deductive arguments, and proofs must be deductive. However, in most other disciplines arguments are largely inductive, so many students may be at first unaware that we are looking for deductive proofs. Students need to understand the difference between observations that they can make from dynamic geometry software [See SLO 6] and deductive arguments that show that something is always true.
Another distinction is often made between synthetic proofs and analytic proofs. Synthetic proofs are those that are made in the style of Euclid, in which we make arguments from axioms without coordinatizing the points, lines, and other geometric objects being studied. If we coordinatize these geometric objects by writing down equations for them, we arrive at analytic geometry. Analytic geometry is covered extensively in algebra and calculus classes, and so most geometry classes focus on synthetic geometry.
It is also important for students to understand in order to prove something is not true, they need to find a counterexample. It can be hard for them to realize that if a proof shows for all cases where the hypothesis is true then the conclusion is also true. However, to show a conditional statement is not true, they just have to find one case where the hypothesis is true but the conclusion is false. Finding counterexamples is important to their development of a robust understanding of proof.
In a GeT course, proof can take several forms. It could be very informal or very formal using a given set of axioms. It could be written in paragraph form or as a two-column proof. Students often enter a GeT course feeling scared of the word “proof” and need support to learn that it just means a convincing argument that something is true. Writing a synthetic proof requires the ability to construct a logical argument in a systematic manner. This skill leads to growth in critical thinking and reasoning. Daily, we encounter situations where critical thinking is needed. Geometry is a great course to use proof to help students build problem-solving skills, which is why proof should be at the heart of the GeT course.
Henderson, D. W. (2006). Alive mathematical reasoning. In R. A. Rosamond & L. Copes [Eds.] Educational Transformations: Changing our lives through mathematics (pp. 247-270). AuthorHouse.
National Governors Association Center for Best Practices [NGA], & Council of Chief State School Officers [CCSSO]. (2010). Common Core State Standards for Mathematics. Available at http://www.corestandards.org/Math/