Mathematical Explanations and Arguments The chapter focused on the fine art of mathematical explanations and arguments. The history of mathematical explanations and arguments is complex because what we currently use as reasoning and proof arises from a methodology from 300 BC that was mostly due to the efforts of Euclid of Alexandria. India and China shadowed the mathematical explanations and arguments asserted by Euclid. Pythagorean Theorem, which means that the sum of the squares of the two sides of a right triangle is equal to the square of third side, is one example of the statement and resolution. Due to the Euclid’s published work, Elements, these arguments and explanations are considered to be resounding demonstrations. If we look at mathematical reasoning and proof from a developmental perspective, we understand that the notion of proof or proving occurs early on in the development of a child. About the age of five children apply the trial and error strategy to prove their solutions. By the time the child is about eight or nine years old and in the third grade, they begin to trust in math processes since they can no longer prove all solutions due to their quantity. According the author there are four varieties of proof. These varieties consist of proof by exhaustions, postulational proof, proof by induction, and proof by contradiction. Although the proof varieties have differences in their structure, they all share three things in common. They are that the proof giver, yourself, to have noted some kind of systematic pattern. Secondly, that you establish a claim about the pattern you have noted. The word conjecture is commonly used to describe this notion in elementary schools. Finally, they require that the proof giver defend the claim in a logical way. Proof by exhaustion refers to finding a solution to a problem by using some kind of systematic problem solving that attempts to find all possible solutions. Postulational proofs are more efficient that proofs by exhaustion because they build upon axioms, definitions, and observed patterning. Postulations proofs move logically from some known facts to some new fact. Proofs by inductions are not as clear cut as they tend to blend with postulational proofs slightly due to the fact that mathematical induction requires apodictic reasoning, which is basically reasoning based on the Greek proof, apodeixis. Induction proofs are especially effective when one needs to form a statement that is true for all whole numbers. Proofs by contradiction establishes the truth of a statement by assuming that the statement is false and deducing a contradiction.
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As an elementary teacher who has not taught math in over 10 years, the reading was quite difficult to comprehend. The connections were not meaningful to me as I have taken high level mathematical coursework as I graduated from college in 1991. The requirements were quite different back then. However, I can see from the real world examples the author of the chapter has used to demonstrate the different types of proofs, proof by exhaustion, postulation proofs, proofs by induction, and proofs by contradiction, that elementary children are practicing proofs unknowingly at a simpler level. Of course, their methods for proving are also not referred to as one of the above mentioned proofs.

The “Write” Way to Mathematical Understanding The authors, Whitin and Whitin, assert that algorithms are an effective way to come solve a problem. However, the authors stress children often do not attain an understanding of why or how the algorithm worked. According to the article, research has shown that if you promote children to write about the steps they took to arrive at a solution is a viable way for children to discover and make sense of how and why the algorithm worked. It is further suggested that educators can use the process of writing about mathematics to make an assessment of their students’