Friday, March 27, 2015

What is best: weakly correct or strongly incorrect?

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Here is a question from the New York Algebra 1 (Common Core) Regents Exam from January 2015. It was worth two points.
This solution is given as a model answer worth 2 points.
 Here is a model answer worth only 1 point.
 Here is a model answer worth 0 points.

Please take note that the sample answer that was incorrect is to be given more credit than an answer that was correct.

I have a serious problem with this as it seems to devalue the correctness of a response while overvaluing an explanation for an incorrect response. Euphemistically speaking, talking the talk seems to be more important that walking the walk.  Please take note that writing "fuction" where the word "function" belongs seems to be totally irrelevant.

This is not to belittle the value of being able to explain a process. Explanations are important. But the hierarchy should be:
1. Correct results well explained.
2. Correct results weakly explained.
3. Correct results not explained.
4. Incorrect results with support.
5. Incorrect results not supported.
Please take note that an incorrect result with support given is simpler to fix that an unsupported incorrect result. Just as a doctor will ask you "where do you hurt?", fixing a solution ought to begin with "where did it go wrong?" That is most easily answered when the result's alleged support is given. 

I refrain from using the concept of "explanation" in relation to an incorrect answer as such answers cannot be explained away. Hopefully they can be repaired.

Along with this dilemma is the misleading description of "function" as stated in the Common Core. Read this:
Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
The words "from" and "to" in the first sentence is misleading. The idea of "with" would be an improvement, as in "A function is a pairing of elements from one set (called the domain) with elements of a second set (called the range) such that each element of the domain is paired with exactly one element of the range."

The description used by the Common Core also gives the impression that a function must have a graph. Try to graph this function: y = the first letter in the word "x", where the domain can be any list of words.  It is a function, but can you graph it?

My favorite example of a function, that I used in class plenty, is the bar code scanner at the local supermarket. To get students to latch on to the notion, I asked them what would have to happen for it NOT to be a function? They quickly agreed that a single item should result in only one price.

Getting back to question 27, is it good to penalize a student for knowing a correct answer to a simple question? I have known many students who interpret "Explain your answer" as if it said "write your answer in sentence-paragraph form."  The answer is so obvious that they do what they think they should and move on.  And they get 0 points?

To put this question in perspective, evaluate all three responses as if they were given to a student by a tutor or teacher as answers when a student posed the question. Is the 1-point answer more valuable then? Naturally the 2-point response is best, but is a supported wrong answer more valuable than an unexplained correct answer? 

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