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If rigor doesn't mean more challenging problems, then what does it mean?

There is a big push for rigor in common core mathematics, but I'm not sure exactly what rigor means (I'm pretty sure it has to do with precision, which would be ironic).

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  • $\begingroup$ I am not sure I agree with the premise "There is a big push for rigor in common core mathematics." I have the CCSS Math Standards in front of me right now, and the word "rigor" does not appear anywhere in the document. $\endgroup$ – mweiss Feb 28 '16 at 2:50
  • $\begingroup$ The comment above is not meant to be construed as a criticism of the Common Core, by the way. I am just wondering why the OP thinks that the CC uses the word "rigor" without defining it -- I don't see it used at all. $\endgroup$ – mweiss Feb 28 '16 at 2:53
  • $\begingroup$ @mweiss: This page here from the official CCSS site identifies it as one of the 3 "key shifts in mathematics" (see #3) -- corestandards.org/other-resources/key-shifts-in-mathematics $\endgroup$ – Daniel R. Collins Feb 28 '16 at 3:44
  • $\begingroup$ One thing I'll point out is that if you Google "common core rigor", almost all of the articles on the subject are written by one person, Barbara R. Blackburn, PhD -- barbarablackburnonline.com/my-team/barbara-blackburn-ph-d $\endgroup$ – Daniel R. Collins Feb 28 '16 at 3:48
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    $\begingroup$ Very interesting. The introduction to the CCSS Standards talks about "focus and coherence" but makes no reference to "rigor". I wonder when the "3 shifts" were written and how they relate to the writing of the Standards themselves. $\endgroup$ – mweiss Feb 28 '16 at 4:40
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Rigor refers to use of the axiomatic method -- being aware of what initial definitions are taken, and being explicit about the chain of logical reasoning that one uses to achieve results.

Wikipedia says this about intellectual rigor in general:

Intellectual rigour is a process of thought which is consistent, does not contain self-contradiction, and takes into account the entire scope of available knowledge on the topic. It actively avoids logical fallacy. Furthermore, it requires a skeptical assessment of the available knowledge. If a topic or case is dealt with in a rigorous way, it means that it is dealt with in a comprehensive, thorough and complete way, leaving no room for inconsistencies.

Scholarly method describes the different approaches or methods which may be taken to apply intellectual rigour on an institutional level to ensure the quality of information published. An example of intellectual rigour assisted by a methodical approach is the scientific method, in which a person will produce a hypothesis based on what they believe to be true, then construct experiments in order to prove that hypothesis wrong. This method, when followed correctly, helps to prevent against circular reasoning and other fallacies which frequently plague conclusions within academia. Other disciplines, such as philosophy and mathematics, employ their own structures to ensure intellectual rigour. Each method requires close attention to criteria for logical consistency, as well as to all relevant evidence and possible differences of interpretation. At an institutional level, Peer review is used to validate intellectual rigour.

And this about mathematical rigor in specific:

Mathematical rigour can be defined as amenability to algorithmic proof checking. Indeed, with the aid of computers, it is possible to check some proofs mechanically.[4] Formal rigour is the introduction of high degrees of completeness by means of a formal language where such proofs can be codified using set theories such as ZFC (see automated theorem proving).

Most mathematical arguments are presented as prototypes of formally rigorous proofs. The reason often cited for this is that completely rigorous proofs, which tend to be longer and more unwieldy, may obscure what is being demonstrated. Steps which are obvious to a human mind may have fairly long formal derivations from the axioms. Under this argument, there is a trade-off between rigour and comprehension. Some argue that the use of formal languages to institute complete mathematical rigour might make theories which are commonly disputed or misinterpreted completely unambiguous by revealing flaws in reasoning.

In the context of Common Core objectives, rigor has been defined as:

Rigor: In major topics, pursue conceptual understanding, procedural skill and fluency, and application with equal intensity.

See the document below for several examples in mathematics education, each of which emphasizes understanding/explaining the reasoning (in addition to computational fluency):

Louisiana Teacher Self-Learning Series

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  • $\begingroup$ hmm, I think I don't like the idea that common core tries to redefine a word that not only has a very specific definition, but means "the value of having specific definitions" $\endgroup$ – Zackkenyon Feb 27 '16 at 16:28
  • $\begingroup$ @Zackkenyon: I agree with that... and you could make a broader complaint that CC does that to a whole lot of terms (I've seen "matrix" get used in a nonstandard way, e.g.). A confounding problem is that elementary teachers are so ignorant of real mathematics, they have no filter or warning system that tells them when the vocabulary has gone off-rails. $\endgroup$ – Daniel R. Collins Feb 27 '16 at 17:30
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Rigor is the standard of mathematical practice, just like experiment is the standard of scientific practice. Rigor is popularly described as that which Euclid did in his Elements. In practice, rigor means being explicit about what you have assumed, what you've demonstrated, and why you believe your demonstration follows only and solely from what you've already assumed.

Rigor, in the mathematical sense, is a distillation of the practice of human arguments. When people argue with each other they have all sorts of tools at their disposal, the two largest collections of which are Logic and Rhetoric. Mathematical arguments are different from arguments in other parts of human activity: they are arguments about whether or not a certain convention which we've agreed upon (the assumptions) is being properly applied in a given situation (the proof).

The standards of these conventions are given by formal systems with axioms and inference rules. These technical terms are often much more frightening than they ever need to be, as they are really just another way of saying "If we all agree to do things in this way by following these rules, then have we followed those rules in this case?"

There are plenty of reasons for why mathematical rigor has developed into the "standard" of axioms and inference rules that we use today, but they can all be traced back to a single human desire: that of certainty, or that of eliminating possible or future doubts. We want to have a feeling that once we make an argument in mathematics we need not ever question its use ever again, and this means eliminating as much as possible any doubts that we've done as we've said we done. This need to be absolutely clear about each and every thing that we've assumed and demonstrated is what has developed into modern formalisms of mathematics using symbolic notation.

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