Zero Is An “Unlike Term” or (7x+4y)/11=???

Part IV of the Zero Saga

If I had been a better math student, I could much earlier have articulated this argument against the use of zero to represent late or missing school work in the mean (or any other) average. helped me clarify my thinking so I could progress from using the nearly useless phrase, “apples and oranges,” to “unlike terms.”

Earlier I mentioned a post on the Teacher Magazine blog that includes some comments that I think are demeaning to the intelligence of teaching professionals. In other words, the support of, and even highlighting, the use of zeros by teachers.

If teachers really understand basic math, it confounds me that they can put zeros in place of missing assessments and then run a mean average on the whole collection of numbers.

First, a number that represents the value of a missing assessment, i.e., zero, has no business in a collection of numbers that represents actual assessment evaluations. The zeros and the grades are “unlike terms” (you learn about this in pre-algebra, so I’d expect that teaching professionals, with college degrees, would know this) and cannot be added together! (The zeros represent work not done, and the grades represent evaluations of student output.) If you can’t add them up legitimately, how can you divide the false sum by the total number of assignments both completed and uncompleted and expect to come up with a number that represents what the student knows about the course goals?

Second, introducing zeros to a collection of assessment evaluation numbers (grades) violates the concept of the mean as a measure of central tendency. By introducing “outliers,” even if they were “like terms,” the mean is corrupted and does not represent a measure of central tendency, or in other words, a number that reflects the main body of the student’s assessment scores.

For more details, refer to a post I made sometime ago on statistical integrity.

Rationalizing about the “real world,” “teaching responsibility,” and other overt attempts at student manipulation won’t make the facts go away. Placing zeros that refer to missing work in collections of evaluative data  is invalid, illogical, and frankly, unethical.

Shift emphasis from attempted (but largely failed) control of student output to evaluation of student achievement relative to district course goals and the picture becomes clearer. In other words, switch from noting “work completed,” to evaluating and noting the achievement of learning goals.

Does that make too much sense?


6 thoughts on “Zero Is An “Unlike Term” or (7x+4y)/11=???

  1. I just had a conversation with a couple of older teachers who were appalled that I refuse to give “0” for missing assignments.

    One actually accused me of giving credit for work not completed. When I explained to her that 50% is still failing no matter what, she still looked at me as if I were a loon.

    Another said to me that she wants to be able to “show” parents how much work their child is missing. I told her that I can show parents missing work as well because ALL missing assignments are still given 50% and any completed work is never given anything below 60%. Of course, she also looked at me like I was crazy.

    We have quite the battle on our hands, don’t we!

    Yes, we do, Ms. Teacher. Stick to your guns! 🙂

    “One actually accused me of giving credit for work not completed.”

    That’s something else I need to write about. Teachers evaluate learning. Schools give credit based on teacher evaluations. Teachers don’t give credit. –Hugh

  2. Each week I give you a set of coins. Each week the value of the set of coins that I give you varies.

    Week 1: Value of coins = $5
    Week 2: Value of coins = $7
    Week 3: Value of coins = $0
    Week 4: Value of coins = $5

    Average value of coins I give you each week = (5+7+0+5)/4=17/4=$4.25

    If a student does not give me the homework that I set, then surely the value of the homework that I recieve is zero.

    (Of course, that only works if you give grades. I dont. I dont see any value in them.)

    Donalbain, I’m happy to hear that you don’t give grades. You must teach in a very progressive school.

    Still, the flaw in you logic is this: grades are alphanumeric representations of teacher assessments of student work. A zero doesn’t represent an assessment, but rather the lack of one. So you cannot combine numbers that represent an evaluation and a numbers that represent the absence of an evaluation, and come up with anything but gibberish that doesn’t reflect achievement accurately.

    On the other hand, numbers lend themselves very well to mean averaging, but numbers have no ability to judge how they should be combined. Keep you eye on what the number represents. That is what determines how you can combine the numbers. –Hugh

  3. I forgot to add something!

    If a homework automatically gets 60%, how does it work if a student always hands in the homework, doesnt get 60% of the questions correct? How can you show improvement in his achievement if he ALWAYS gets 60%?

    An artificial (unlikely) possibility — and this is a common logical error — is often used to derail a sound argument. But it remains unlikely and artificial, and we make the mistake of accommodating the unsupported hypothesis while failing to address the greater problems the sound argument may solve, e.g., the “show improvement” argument above, which is offered as a reason for continuing the use of zeros, is beside the point that education literature supports the prohibition of the use of zeros (O’Connor, Marzano, Guskey, Wiggins, Stiggins, et al).

    But people bite on it and swallow it all the time! In logic, it’s referred to as an “irrelevancy.” (Learn more about irrelevancy in Robert Gula’s book Nonsense: A Handbook of Logical Fallacies.)

    And if 60% doesn’t melt your butter, use 50%. Just don’t fall into the trap of using the zero. –Hugh

  4. When I had absent students, I would go ahead and put in a zero in my computer gradebook. If they neglected to get their makeup work, their weekly printout would show their average with that zero figured in. I would then show them what their grade would be if they turned it in and just made the lowest C possible. I had a pretty good success rate on getting makeup work with this. However, I do think that it would have been a much better representation of what a kid knew if I had put in a 50% just before “official” grades were recorded. I also would’ve saved myself some headaches. Wish I could go back and change it now.

    Eiela, nobody who has been teaching for any length of time is totally innocent of grading abuse. We didn’t know what we didn’t know, and nobody was teaching us any differently. I’m glad that you can think about it, and I’m sure that reflection will influence how you grade future students.

    I believe that, in the absence of grading instruction, teachers adopt the grading methods of others and adapt them to their own personality. I could be full of beans, but have you ever looked around at your peers and noticed that language arts teachers, followed by math teachers are the most “strict” graders, while social studies teachers, on the whole, tend to be more “laid back” about the issue?

    I have no data to support those ideas, just years of observation. 😉 –Hugh

  5. I would just like to correct a misunderstanding. My “show improvement” question was not meant to be a defence of using zeroes. It was meant to be a criticism of giving a set grade for anything that is handed in.

    My grading system is as follows (for those who care about such things)..

    I only ever put comments in the students books. My reading of research (I cant recall titles at 12:30am!) is that comments and grades combined mean that only the grade is looked at, while my experience is that grades on their own do not give useful formative feedback.
    Each reporting period I am expected to give a National Curriculum level as well as an effort grade and a written comment. The NC level is based on standards, and the effort grade is down to me. In addition I give each child a % grade for number of homeworks done.

    Thanks for following up, Donalbain. My apologies for misinterpreting your meaning.

    Sounds like you are where I’d like my district to be in about a decade. Keep up the strong work.

    BTW, I have a feeling that you’re not in the US, based on your spelling of the word “defence.” If I’m not prying too deeply, which continent are you writing from? –Hugh

  6. I am writing from the United Kingdom. What I described may not be universal, but it is fairly common. I am only a newbie teacher, and it is how we were encouraged to do things during our training.

    p.s. I did not know that Americans spelt defence differently. Every day is a schoolday!

    How right you are. Welcome aboard. 🙂 –Hugh

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