Saturday, January 3, 2015

Favorite comments of '14. cont: Anonymous, Barry Garelick, and SteveH

On More Common Core-inspired issues: the communication skills of non-native English speakers... and of Common Core Authors:

Anonymous said...
What I don't understand is why, instead of having students verbally show that they understand, can't they design problems that can't be solved unless the student has conceptual understanding? The Singapore math books are full of such problems--and interestingly, back when they were first developed, a huge number of the students using them were English language learners.
Barry Garelick said...
Anonymous: Depends what you mean by conceptual understanding. A problem that asks for how many 2/3 oz servings are in 1 3/4 oz of yogurt requires that a student recognize that the problem is solved by division as well as a procedural knowledge of how to do fractional division. If the student cannot explain why the "invert and multiply" rule works but shows he understands what the problem is asking and how to solve it, does that mean he/she lacks conceptual understanding?

I agree with Katharine that McCallum and others should be making more public statements about what CC math standards require and what they do not. After I wrote my first article offering alternative interpretations of CC math standards (in Heartlander), McCallum showed some interest and even blogged about it. He even acknowledged that the standard algorithm for multidigit addition and subtraction can be taught in grades earlier than 4th--the grade in which that algorithm is mentioned in the CC standards: "By the way, the standard algorithms for addition and subtraction are “strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.” The standards do not forbid them being introduced in Grade 2—nor do they require that."
  
SteveH said...
Conceptual understanding is just that - conceptual. Too many educators mistake it for proper mathematical understanding. Conceptual is at the level of motivating why you are learning the material in the first place. It's at the pie chart level, and that simple kind of understanding of fractions provides no support for understanding rational fractions. Many teachers say that the toughest math class in high school is Algebra II because that's when those conceptual understandings fall apart. Understanding should be built on identities, not pies or bars.

Some students need more motivation than others, but it is neither necessary or sufficient for real understanding. That is built from doing and understanding nightly individual problem sets.

As for testing, how can standardized tests ever test conceptual understanding or problem solving in the sense of applying mathematical ability to new problems - ones not covered in all of the problem variations encountered in homework?

What are teachers, potted plants? Are they incapable of making those judgments even with seeing the kids every day? What is the purpose of yearly standardized tests? Are they used because we can't trust teachers or are they used just as a safety net? If they are used as a safety net, then it would be much simpler to just test the basics - results that can give specific things to correct. As it is now with NCLB, our schools get vague scores on "problem solving" and no other details on why the results aren't as good this year. Shouldn't they already know what the problems are?

Why on earth would anyone expect a problem solving yearly standardized test be part of a critical thinking feedback teaching loop?

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