Starter Set for Math Curriculum or Year Plan for
Reasoning and Proof

Concepts or Big Ideas Outcomes Activity Sequence Evaluation
levels
Reasoning - begins with examination, comparison, and evaluation. Examination, which can include discovering similarities and differences to organize, categorize, seriate, make connections, create correspondences and create other structures. All which can be connected for deeper understanding, to solve problems, and generalize to infinity.
  • Recognize reasoning and proof as fundamental aspects of mathematics;
  • Use data to support mathematical ideas.
  • Asks others to explain their reasoning.
  • Desires to explain their understanding, reasoning, and procedures.
  • Rejects solutions that have insufficient evidence and lack reasonable explanations.
   Exemplary - Students at this level provide arguments that demonstrate a conjecture or statement is true for all cases. They identify any assumptions or givens, a sequence of deductions used to build an argument, and finally a concluding general statement (general for all cases - infinite).

Progressing - Students at this level recognize a need to provide a general argument, which includes all possible cases, however, their argument is incorrect or incomplete mathematically or logically, to make it unacceptable in some manner. One possible example at this level is using an inductive reasoning or empirical evidence to suggest a proof by giving several accurate examples to support a conjecture, but describing it as an error of proof because it is not possible to prove all cases (general argument) in this manner.

Beginning - Students at this level recognize a need to provide mathematical justifications. However, their justifications are specific (empirical) not general. This level could be further divided into those who check only one case or multiple cases, or those who check multiple examples. The inclusion of multiple cases could also be divided into categories by how the multiple categories are selected. Systematically selected to check essential representative cases (all possible combinations of say addition and subtraction of odd and even numbers as (even and even, odd and odd, and even and odd numbers), or selected to check extreme cases, or randomly selected cases, or selected with consideration of generic classes.

Not yet - Do not provide a mathematical explanation to describe a valid conjecture. May accept a conjecture as true because a teacher, parent, or text "says" it's true or says a conjecture is true without any reason to support it. (Sum of two odd numbers is even, because it is. Or use circular reasoning. The sum of two odd numbers is always equal because they will always be equal).

 

 
Conjecture is an idea or conclusion based on insufficient evidence and are created without sufficient evidence or reason to prove. Therefore, are often condidered as a starting point for a proof.
  • Make and investigate mathematical conjectures;
  • Makes generalized statements from specific examples.
 
Proof - Every idea in mathematics has to be proved before it can be used as a mathematical idea. A mathematical argument that is used to prove a conjecture must have a valid reason for every change (or every step) for what is in question to something that is known or has previously been proven mathematically.
  • Explains how a problem was solved.
  • Interprets and explains other students' solutions as accurate, inaccurate, or not sure.
  • Looks for addition examples for justification.
  • Solves problems in different ways to justify accuracy.
  • Recognize that a proof has to justify every possible example.
  • Uses concrete representations to justify all examples of an argument and proof.
Develop and evaluate mathematical arguments and proofs for the following conjectures:
Abstract proof - Mathematical ideas can be proven by manipulating an abstract representation, in symbols or as a model, based on previously proved manipulations, to prove a conjecture.
  • Use symbols to represent all possibilities when proving a conjecture.
  • Uses models to demonstrate and justify reasoning.
 

 

Dr. Robert Sweetland's notes