Sunday, November 15, 2015

Explaining Your Math: highly controversial!

There are now over 450 comments on Barry Garelick and my article in the online Atlantic, Explaining Your Math: Unnecessary at Best, Encumbering at Worst.

Apparently, this is a rather controversial topic. Critical comments appear to boil down to 7 categories:

--What the Common Core actually says: Some people stated that there’s nothing in the Common Core itself that requires students to explain answers and thought processes verbally and diagrammatically. (We noted that there are parts of the Common Core that nonetheless can be, and are being, interpreted in this way).

--Our specific examples: Some claimed that examples we used to illustrate work showing aren’t representative of what’s generally going on, and that good teachers would be more flexible and reasonable about what constitutes adequate explanations. (That would be nice, if so.)

--Who is actually affected: Some claimed that our objections apply only to very small subsets of kids. (We pointed out that these practices are problematic for all students, and in particular for second language learners and children with language impairments).

--The virtues of showing your work: Some people conflated showing work (which we agree is reasonable wherever there’s work to show) with explaining answers and thought processes verbally and diagrammatically.

--The virtues of doing math proofs: Some people conflated doing math proofs (which we agree there should be more of in high school math) with explaining answers and thought processes verbally and diagrammatically. (We pointed out that there is actually relatively little emphasis on mathematic proofs in both the various Common Core-inspired curricula and tests).

--Communication skills necessary for math-related professions: Some people believe that having students provide the sorts of verbal and diagrammatic explanations we critique in our article will help prepare future engineers and scientists for the communicative demands of their jobs. (The question then is whether engineers and scientists who learned math in pre-answer-explaining times are deficient in their communication skills compared with their more contemporary counterparts.)

--Counter-exemplary anecdotes: Some people described how well explaining answers and diagramming thought processes work for their students or kids.

--Faith: Some people are sure that meta-cognitive processes are the best way to develop conceptual understanding. (We would say that a better way is to emulate the countries that outcompete us in math, giving kids more direct instruction and individualized practice in conceptually challenging math problems).


Marie said...

Interesting! How many of each? (said the right-brained mom...)

Barry Garelick said...

Regarding the issue that CC doesn't require explanations, etc. We quote something that appears on the CC website, that calls for students to justify their answers. Website is not the standards, true, but still... Then, the content standards themselves contain the phrases "students shall explain.." and "student shall understand that..." . The intro to CC standards states that whenever "understand" appears in the content standards, to link that with the Standards for Mathematical Practice, the first of which is "Make Sense of Problems and Perservere in Solving Them". That SMP states in part:

"Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches."

Such language has served as gasoline thrown on the fire of math reform that has been burning for many years, and which has fetishized conceptual understanding.

Anonymous said...

It seems like two of those subgroups, (the virtues of showing your work, and the virtues of doing math proofs,) are quickly convinced that explaining your math Common-Core style is bad.

It seems like two of those subgroups, (the specific examples, and what Common Core actually says), also agree that explaining your math in essays is bad.

There are only three subgroups that seem to think this is a good idea: the communication skills necessary for math professions, the counter-example anecdotes, and the metacognition/faith group.

So it sounds like many of the supporters of "explaining your work" don't actually support it.