I propose here ideas teachers need to know and pay attention to when teaching mathematical proofs and how to prove.
A. What is a (mathematical) proof?
I define proof as a relational network of claims (propositions and conclusions), substantiation (established knowledge that makes the claim legitimate) and appropriate connectives so sequenced to justify why the conclusion is a logical consequence of the premises.
In mathematics teaching, a proof is made to establish the generality of a statement (most times, they are theorems) and of course to initiate the students to this kind of mathematical practice that is unique to mathematics.
Proving is the activity of constructing a proof. It is a creative reasoning process to build up substantiated argument.
What a proof is and how to prove are difficult to learn and are difficult to teach. Note that ‘difficult’ does not mean not doable.
B. What are students’ difficulties with proofs?
Students’ difficulties with proofs and proving involved (1) reading and understanding proofs, (3) evaluating the suitability of proof and (3) writing a deductive proof. So, basically any mathematical task that involves proof is hard for learners.
In addition, research has shown that when asked to prove a mathematical statement or verify the correctness of proof students tend to use empirical data or specific cases. They oftentimes do not know where to start or unable to use existing knowledge strategically.
C. What to pay attention to when teaching proofs and proving
If students are to gain an understanding of, and a capability with, how to construct proofs, the teaching need to pay attention to creating opportunities for students to understand and appreciate the structure of the proof. The structure of the proof includes the claims, the substantiation for the claims and the logical connectives that helps in the logical sequencing of the statements. Of course the claims should be mathematical and the substantiation are accepted mathematical principles, definitions, properties, theorems, previously proved statements, etc. How can we foreground these in our lessons on proving? Here are two complementary techniques
The flowchart (see Figure) helps show the structures. The method was proposed by Miyazaki, Fujita and Jones (2015). I invite you to read their paper on Flow-chart proofs with open problems as scaffolds for learning about geometrical proofs published in the International Journal in Mathematics Education.
Naming (word use)
This strategy is based on the idea of mediating knowledge through words and word use. It nicely complements the flowchart method. Here you pay attention to the use of words both to communicate what exactly is being done and why. And not just pay attention but say or write the words. When it is a claim, you need to say so. When it is a substantiation for the claim you need to indicate it with words such as because, since, etc.This can be done by highlighting them when you write the proof and requiring learners to use these words when they make statements orally. For example we should write/say “We can claim that ΔCAD is isosceles because it’s two sides have the same length being radii of the circle”. It is also important to be explicit with your connectives. For example, “Let q be the measure of ∠CBD. If this is the case, then ∠ADB = q also since the base angles of any isosceles triangle have equal measures.
Making explicit the claim, substantiation and logical connectives in the proof helps students see the structure of the proof. You can find an example of these in How to Prove the Central Angle – Inscribed Angle Theorem.
Feel free to share your techniques proofs in the comment section.