Barry Garelick and I have a piece up on Education News.

Some excerpts:

At a middle school in California, the state testing in math was underway via the Smarter Balanced Assessment Consortium (SBAC) exam. A girl pointed to the problem on the computer screen and asked “What do I do?” The proctor read the instructions for the problem and told the student: “You need to explain how you got your answer.”

The girl threw her arms up in frustration and said “Why can’t I just do the problem, enter the answer and be done with it?”

…

[For some problems] the amount of work required for explanation turns a straightforward problem into a long managerial task that is concerned more with pedagogy than with content. While drawing diagrams or pictures may help some students learn how to solve problems, for others it is unnecessary and tedious.

...

Is it really the case that the non-linguistically inclined student who progresses through math with correct but unexplained answers—from multi-digit arithmetic through to multi-variable calculus— doesn’t understand the underlying math? Or that the mathematician with the Asperger’s personality, doing things headily but not orally, is advancing the frontiers of his field in a zombie-like stupor?

Or is it possible that the ability to explain one’s answers verbally, while sometimes a sufficient criterion for proving understanding, is not, in fact, a necessary one?

## 7 comments:

English class: explain Captain Ahab's obsession with the whale, make sure to use at least one text-to-self and one text-to-text reference, also use math to explain your answer, the math must include diagrams of how it relates to the book.

Ha Ha! Auntie Ann, I was just about to suggest a similar scenario. That is not to say that once in a while, asking students to describe how they arrived at a math problem answer using sentences might not be appropriate. But only once in a while, because crafting sentences to explain procedures that can be illustrated with mathematical expressions is hugely more time consuming and therefore frustrating. Way to make kids hate math.

"Or is it possible that the ability to explain one’s answers verbally, while sometimes a sufficient criterion for proving understanding, is not, in fact, a necessary one?"

Explaining in words is neither sufficient or necessary. Are you going to flunk a student who gets the correct answers but does not explain them well?

"They" want you to explain each problem with words. What does that imply? That is the whole point of giving many problems on a test - to test understanding and flexibility on a wide variety of problems. Doing the problem, showing your work, and getting the correct answer is all that a teacher needs to determine sufficient understanding. The work shows the understanding, not words.

Back when I taught math, words might get some partial credit, but I always could tell by examining their math - scribbles and attempts to solve on the paper. If I could see their work and they got the answer correct, then no words were necessary.

Nobody can do well in math with just rote understanding. This is a huge fallacy. One can have an inflexible understanding or one with gaps and holes, but there are no words that fix that. It takes more practice and proof by doing and getting the correct answers.

Teaching math is all about how words are not enough. Group work in class is all about group words. What individual students need for understanding is to do the homework sets individually and consistently get the correct answer without help from a teacher, tutor, or group.

Giving many problems and demanding wordy answers on a test are mutually exclusive. In the time it takes to explain in words one problem, a student could demonstrate their proficiency on several problems with different mathematical concepts. Writing wordy explanations is much slower than giving a student a variety of different questions.

"Explaining in words is neither sufficient or necessary. Are you going to flunk a student who gets the correct answers but does not explain them well?"

No, but I might

notflunk a student who gets the wrong answer (for example, due to a stupid arithmetic mistake) but can explain how to solve the problem.Exactly. That's what I did when I taught math. Sometimes a student would get stuck early in a problem, but generally know how to solve the problem and would explain that in words. I gave them more partial credit, but not full credit.

If they developed the problem through to near the end, but got stuck, I could tell whether it was an understanding issue or a "typo" sort of mistake. Perhaps they missed a minus sign. I could always tell what level of understanding they had. Words are not necessary.

I always told my students to show their work. If they didn't, then they would not get partial credit because I could not guess what was going on in their heads. They could do this with words, but the math would get them more partial credit. Words are vague, but the math is not.

My nephew is in the habit of erasing his mistakes. I keep telling him to just cross them out and start again, so a teacher can see if you were on the right track. But, he's a neat freak and hates the messy paper.

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