Tuesday, March 28, 2006

There's this line that you cross in teaching,

when you stop thanking the good lord for any break from the usual routine, any day away from the evil monsters students, and realize you'd rather be in your classroom than anywhere else (and not just to keep the kids from trashing the room). Some of the math teachers at the training today were still on the other side of that line, secretly happy to get a few hours out of their classrooms and a short week. I would much, much rather have been teaching. Plus, five of the 14 teachers at my school were out for training today, not a recipe for rigorous instruction...

The trainers made it clear from the get-go that we weren't there to debate the merits of the scoring rubrics or the questions, just to learn how to apply the rubrics consistently in order to ensure fair scoring according to the state's answer key. We were really good for the first hour or so, as we watched a video of a chirpy blond teenager woman reading aloud from the scoring guide. As time wore on, though, we began to have questions. What if a student wrote...? Why did that get a 1 (instead of a zero, or a two, or a three)? Isn't that double jeopardy? (getting penalized twice in the same question for the same conceptual error) And pretty soon, these questions edged dangerously close to debating the merits of the rubric.

The scoring rubric is a fascinating mix of questions that are scored so loosely as to render the mathematics all but meaningless, in my opinion, and questions that are graded with draconian pickiness about details that seem all but meaningless (also in my opinion). An example of the latter is a question like this (no, I am not quoting the exact question):

On Tuesday, Julie saw a certain number of elephants, e. On Wednesday, she saw 5 more than 3 times the number of elephants she saw on Tuesday. Write an expression for the number of elephants Julie saw on Wednesday.

The right answer, of course, is something resembling 3e+5. The children lose a point automatically if they include an equals sign, because that makes it an equation, not an expression. I understand that as they move towards algebra, the difference between an expression and an equation will become more and more important, and therefore am willing to concede that the rubric is probably fair.... but many math teachers in the room were up in arms about this question. And consider the kid who writes 3e+5= and doesn't put anything on the other side of the equal sign. Or the kid who writes w=3e+5, introducing a variable to represent Wednesday's elephants. Sure, it's an equation, but is it wrong?

Interestingly, the very next question on the test went something like this,

Gwendolyn is comparing two expressions. The first expression is 5^3. The second expression is 8^2. Which expression is greater?

Acceptable answers included both 5^3 and 125. Can an expression be an integer, alone? I don't actually know the answer to this question.

Here's another interesting math fact that I learned from the grading training:

When a child writes something like 3*6=18*2=36, showing each step of a sequence of operations (compare to 3*6*2=36), that's called a "string" and is not considered acceptable in 6th grade. I'm assuming this is because it is a sloppy habit that could make algebra more difficult, although many adults use strings as shorthand (it's easier than 3*6=18 and 18*2=36, which would be okay). Never mind that we understand the child's reasoning perfectly, we can't count it as acceptable showing of work.

Okay, fine. I'll grant you that one. A few questions later, we see a child who has multiplied numbers in sequence, a la


Isn't that a string? I ask, raising my hand. No, that's not a string. A string has equal signs. But isn't the line at the bottom of the multiplication problem essentially an equal sign? I mutter to myself, knowing this is not the time for arguing the meaning of mathematical terms. I just hope the sixth graders have been informed of the subtleties of using and avoiding strings...

But this is my favorite:

Solve the equation for m. m-5=12 Show your work.

Now, when you're eleven, or really any age, there are two obvious ways to solve this problem. The first is to use inverse operations, and add five to both sides. The other is guess-and-check.

Remember "guess-and-check" -- try out some numbers until you find one that makes the problem correct? (One of my own math teachers jokingly referred to it as "search and destroy"...).

Guess-and-check is an acceptable strategy for solving this problem - if and only if the student shows evidence of having tried at least three numbers before arriving at the correct answer!

So the student who writes
and then puts 17 on the answer line
gets only partial credit,
because he or she did not show evidence of using inverse operations, and did not show evidence of guessing three times before arriving at the answer.

Outcry: What about the lucky guesser?!

If the kid happens to guess right on the first or second guess, he or she needs to make up additional wrong guesses in order to have three guesses altogether. Really. I'm serious.


As my friend put it, "That's three hours of my life I won't get back..."


Blogger -llm. said...

As a 4th grade math teacher and a math coach for 4-6th grades, my response is AHHHHHHHHHHHHHHH!

There. I feel better now.

9:56 AM  
Blogger Amerloc said...


11:32 AM  
Blogger Ms. Teacher said...

That is unbelievable. How very frustrating.

4:03 PM  
Blogger Mr. Person said...


There is a reason for encouraging students not to write strings, but I wouldn't see any point in judging a student negatively because he or she thought in strings.

The reason is pretty straightforward. Students will be writing something that isn't true: 3 * 6 = 18 * 2, to use your example. The last part, 18 * 2 = 36, is true, but the first part isn't, nor is 3 * 6 = 36, which is also part of it.

The vertical work is not standard mathematical writing; it is a "workspace," so I can understand why the rule wouldn't apply there.

Skipping my way around through your post here, the answer to your question about expressions is, yup. To put it too simply, 2 * 2 and 4 are both expressions. The latter is in a simpler form, because it has fewer terms.

Also, the phrase "which expression is greater" is bad bad. To be absolutely accurate, you don't compare "expressions." You compare the "values" of those expressions.

And, to be fair, a student who writes w = 3e + 5 HAS written the correct expression--and has couched it within an equation.

4:09 PM  
Blogger pseudostoops said...

you could even argue that the kid who guesses and checks correctly on the first try isn't a lucky guesser, but a capable estimator. and estimating was a big math standard in california when i was teaching there. argh.

4:51 PM  
Blogger Jules the Crazy said...

...And thus the stupidity of mass-market open-ended math exams!

I am disgusted, though not terribly surprised, about that last one. needing at least three guesses? that is just inanity at its best. pity the smart kids!

7:04 PM  
Anonymous Bill said...

Yeah well...on question 28 of the 6th grade exam, students are asked to draw a rectangle with a perimeter of 34 units. A grid is provided. Easy enough. Then students are told to write the values for each side of the rectangle. Obviously there are many different rectangles a student can make. Many students went about answering the question just like they would have in a classroom context. 10+7+10+7=34 units. Well guess what? Students who answered in such a way got half the problem wrong. They were to simply write 10, 7, 10, 7. It was obvious after grading a few tests that a majority of students were losing points needlessly. Yet, the state made a point to say that we were to grade the exams using a holistic approach. I wonder how many students will barely fail and have to needlessly go to summer school.

7:29 PM  
Blogger ms. frizzle said...

Hmm, according to my scoring training, that answer would have been acceptable... I was trained for the sixth grade test, as well.

8:58 PM  
Anonymous Anonymous said...

It is almost as if they *want*
to encourage parents to homeschool.


9:52 PM  

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