NY State backs down on inverse flub; no geometry gaffes until later today
New York State asked high school students to find the inverse function of a non-invertible function. 32. If find . Youch. Teachers across the state yelled and screamed, but most importantly called the testing hotline (I never got through, all busy). What was the response?
They backed down. But tried to weasel out and claim they had not made an error. Here is the language in their clarification memo:
“Because of variations in the use of notation throughout New York State, a revised rubric for Question 32 has been provided. Please rescore all students’ responses to Question 32….” And then they allow or or or “An explanation stating that the original function is not one-to-one and therefore there is no inverse”
Surrender? No. They ascribe their error to “variations in notation.” Actually, even if they were right, they have some explaining. How could the test-makers not be aware of “variations in notation”? Also, notice how there is an asymmetric justification burden on a kid claiming (correctly) that the inverse does not exist.
Before we got that memo, what response did they give teachers?
Tuesday late afternoon (report is by a math teacher from the Association of NY State Mathematics Teachers listserve): I called the state today about question #32. I told the woman that by definition, for a function to be invertible it must be 1-1. Her response was that every function has an inverse, it just that every inverse is not necessarily a function. She said that the inverse could just be a relation. I told her that she was wrong and then quoted her a definition that says that a function is invertible if and only if it for every input there is exactly one corresponding output value. I then said that it was only invertible only if we restrict the domain to all value greater than or equal to zero, or all values less than or equal to zero. It the domain is restricted in this way, the student shouldn’t write positive or negative. She said that they can receive full credit if they only stated the positive solution, given that they explained that they restricted the domain to make it 1-1. I asked her why did I have to expect so much from the kids when a bunch of adults who made the exam, didn’t make that distinction? She told me that i wasn’t going to win the argument. This error is more apparent given that the question uses function notation.
Wednesday morning (report from another teacher on the list): I just got off the phone with a person from State Ed (who refused to give his name). … After much discussion, I pretty much got him to agree that the inverse should be a function. When I insisted that the answer should be correct with +sqrt, – sqrt as well as both, he finally said “use your professional judgement!” I then said- why don’t they admit the question was flawed and issue a correction statement so we are all grading the question the same way? His response was we will look into it.
Then Wednesday late morning they started telling teachers that they were working on a memo, which finally came at 1:48 PM.