How would you have graded this simple test?

[mpc]

My reply pretty much agrees with what Richard posted earlier:
That quiz shows a "common mode failure" in the less-than/greater-than/equal-to section. It appears (to me at least) the quiz was intended to test the student's comprehension of what various numbers mean. Just what is "3" compared to "7" for example. There were a handful of basic addition problems to demonstrate "math" ability with the numbers, then a whole slew of the less-than/greater-than/equal-to problems. A single miss-understanding of the ">" and "<" symbols though totally destroyed the intent of that section: does the student understand the numbers. The test answers indicated "no, the student doesn't comprehend numbers at all" which is not the case - most folks reading this thread have commented it is clearly a problem with the symbol comprehension - thus the "come see me" suggestions.

Instead of so many similar less-than/greater-than/equal-to problems - which leads to the "common mode failure" - a test with only 3 or 4 of a particular type of problems and instead a lot more varied types problems would have probably have resulted in a far more representative testing of this student's math/numbers comprehension. Testing ONE method over and over won't help determine what a student did or did not learn - if you test the thing he "didn't get" then it appears he didn't learn anything. It's the testing equivalent to beating a dead horse. This test was graded fairly... but I think it was a poorly designed test thanks to a common mode failure.

Understanding the "why" is often more important than "what" in events: why did the bridge fall down, why did the student miss every single question in the less-than/greater-than/equal-to portion of the quiz? With multiple types of questions, a teacher/student/involved parent can isolate what is the weak spot in a student's understanding. With big class sizes today, limited time, and parents that expect the schools to do everything, unfortunately students rarely receive the analysis of where their weak understanding spots are and what they need to focus on. A friend of mine, with a struggling junior high school aged son, was an example of this. The son struggled in just a few areas in school... during a semester his mother spent the money to send him to the Sylvan program. They took the time to figure out just where his understanding was lacking and worked on that. One semester of weekend classes made all the difference in the world.

mpc

[dkerfoot]

Since the U and the VG are just symbols like the < and >, you really shouldn't care which one is used, so long as you get the concept that it is better to get answers correct. Right?

[cgallery]

Since the U and the VG are just symbols like the < and >, you really shouldn't care which one is used, so long as you get the concept that it is better to get answers correct. Right?

They're both (concept and correct answer) important, right?

Otherwise, why ask kids to show their work?

Otherwise, why give partial credit where there is a slight computational error, but the work shows the pupil was on the right track?

Otherwise why not hand every kid a calculator and tell them "all we care about is the answer, and this is the fastest way to get there?"

[Alex Franke]

I just heard back from a friend on this, and her response was a bit different so I though I'd share it. She's a middle school math teacher, and I posed the question out of context this time. (I'm guessing that "S" means "satisfactory.")

She said: I would probably do an oral exam to be sure he understands the concept of the inequality symbols ... to find out if he truly has no clue, or just got the symbols mixed up because he doesn't know the words for the pictures. For me the symbols would not be the point of the lesson ... probably record an S, but with a note, and spend a little extra time driving it home.

Since the U and the VG are just symbols like the < and >, you really shouldn't care which one is used, so long as you get the concept that it is better to get answers correct. Right?

Maybe he shouldn't, but a kid probably would care if he does understand what U and VG mean. It seems to me that you could clear up the confusion and "make it stick" -- especially with a kid who's not used to bad grades -- without whacking him over the head with it. I think this is the power of the "see me" solution...

[dkerfoot]

They're both (concept and correct answer) important, right?

Otherwise, why ask kids to show their work?

Otherwise, why give partial credit where there is a slight computational error, but the work shows the pupil was on the right track?

Otherwise why not hand every kid a calculator and tell them "all we care about is the answer, and this is the fastest way to get there?"

Sorry CGallery - I forgot to add the sarcasm/irony emoticon...

[LCHIEN]

this thread has probably gone on enough. But as a final thought, there's three things on the test... some addition, some mutli-digit addition and some <=> relationship stuff.

Tests not only test what an individual knws but also how the class in general has picked up the knowlege. And identifies who needs help.

In this case by careful examination she may have determined the "student" knew the relationships but was confused between the two similar symbols. OTOH she may have missed that and was just grading the results (maybe her sister the english teacher was helping grade that evening.)

IN any event there would have been at least a dozen other "narrow scope" tests and such an anomaly would have worked it way out by the law of averages if the student was really any good.

And for all we know maybe the teacher did see the student to explain that he swapped sybols, or maybe he was smart enough to figure it out on his own or his parents in reviewing the tests told him (you guys do look at your kids work, don't you?)

[phrog]

It's the testing equivalent to beating a dead horse. This test was graded fairly... but I think it was a poorly designed test thanks to a common mode failure.

Understanding the "why" is often more important than "what" in events:

mpc

"Beating a dead horse" - That's what I was trying to say. Wish I had said it like that.
Richard

Edit: Can you believe there are so many opinions on a decades-old arithmetic test?

[LarryG]

I concur on the dead horse stuff, but having thought about this overnight, two brief, final thoughts:

1. That the test included more comparison problems than was really necessary is an excellent point. There were, what, fifteen of them? A half-dozen would have been enough.

However ...

2. The inclusion of the two "=" problems (12 __ 12 and 4 __ 4) reinforces my contention that this test was as much about understanding the proper use of the three symbols as it was about understanding the underlying mathematical concepts. It is obvious that the numbers on each side are the same, therefore the only thing being tested by these two questions is, "What is the correct symbol to describe this relationship?"

[Alex Franke]

That quiz shows a "common mode failure" in the less-than/greater-than/equal-to section. [snip] A single miss-understanding of the ">" and "<" symbols though totally destroyed the intent of that section: does the student understand the numbers.

This is really interesting. It seems like you could grade the test a completely different way that related directly back to the goals (whatever those might have been).

- Did he understand how to add larger numbers? Yes (top part)
- Did he understand the concept of equality/inequality? Yes (= versus <>)
- Can he tell when one value is bigger than another? Yes (mechanics)
- Does he understand the comparison symbols? No (symbolism)

...so score would be 75% graded like this, but it certainly more subjective. For example, if the wrong answers weren't so consistent, the teacher might not know how to answer one of them.

So instead of "beating the dead horse," it might have been better to design the test with 2-3 questions that relate back to each specific goal, right?

Edit: Can you believe there are so many opinions on a decades-old arithmetic test?

No kidding! I thought it was interesting, but I didn't know people would be so passionate about it! I think it's great.

[Daryl]

The first thought that came to my mind was dyslexia, but they might not of had that back then. The second thought was, "My long lost brother in arms at math", I stunk.