In a lot of ways, it’s much easier for me to come up with helpful feedback to give on rich, juicy problems (see here) than it is for your typical quiz or test. I find it much harder to think about how to give feedback that helps a kid’s learning when (a) the quiz is full of non-open questions and (b) the kid’s solutions don’t show a lot of thinking. But a lot of classroom assessments end up like that, and it’s important to figure out how to deal with those tough situations effectively.

So: What would you write as feedback on this quiz?

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Some constraints/notes, that you should feel free to reject or challenge:

  1. Assume that we’re dealing with written feedback here. Not a conversation.
  2. Assume that we don’t have to write a grade on this piece of work. (If we wrote a grade on here, some research indicates that would ruin any feedback we gave.)
  3. You might decide to give feedback on every question of this quiz, you might not.

I’ll jump in with my thoughts in the comments. Here are some questions about your choices that I’m wondering about:

  • Would you choose to mark the questions as right/wrong?
  • Would you try to find something to value about this kid’s work in your comments, or will you be all hardass instead?
  • Would you ask questions or give suggestions?
  • Would you write one, several, or many comments?
  • Would you reject the constraints in some way?
  • Would you ask the kid to explain himself?

Excited to read your thoughts!

  • It’s hard to give useful feedback because we don’t know what the student is thinking. Would it be helpful to refer the student to similar worked examples in the textbook?

    • mpershan

      You’re so right that knowing what a kid is thinking allows for really great feedback. We can figure out what misconceptions they have, why they said what they did, etc.

      But I’m not so sure that we can’t help. After all, no matter why this kid wrote what he did, we know two things: (1) this kid got a wrong answer and (2) he didn’t go through a verification process.

      I think that’s enough to get him started on. It’s productive to try to verify your answer, and we can help set him on that path. Then, we can get him back in problem-solving mode again so that we can have better access to his thinking. Maybe a friend helps him, maybe he helps himself, or maybe he calls us over for some help. But I think that the whole thing goes more efficiently if we get him back in the swing of things.

      (It also allows us to give feedback on the task, instead of on his thinking. I think we have good reason to want to do that.)

  • mpershan

    After trying to cajole people into commenting here before I did on twitter (fail!) I was pressured into commenting first. Obviously, this is a tough case. Without too much confidence, here’s what I would do:

    First, I wouldn’t give feedback on the entire quiz. Commenting on everything would give this kid heartburn and too many choices about where to go next. I also wouldn’t mark whether the questions are right or wrong, because I think that would distract the kid from my comment without helping their learning in any way.

    Then, I would instead pick one area that I want this kid to work on, and I’d tell them (a) something that I valued and (b) something productive that I’d like them to think about and work on.

    I’d pick the inequalities (Q2 and Q3) for no particular reason. And here’s the comment that I’d write:

    Good job coming up with potential solutions to Q2 and Q3. Having something to test is important. How can we be sure that these potential solutions work, though? Try swapping in 3 for ‘x’ and seeing if you end up with a statement that is true. If the solution makes a true sentence, try to find others that work. If the solution doesn’t work, can you find some numbers that do?

    I’m obviously trying very hard to give this kid a productive next step. I’m also being careful not to say “your answer” or “your solution,” instead using language that attempts to distance the wrong idea from the kid with “potential solution” and “the solution.”

    Finally, I chose to respect my own constraint about keeping the feedback written. While there are obviously benefits to actually having a conversation with the kid, we need to have ways to productively move a kid like this forward in writing. Who says there’ll be time for a full conversation?

    • Cal

      Judging by their brief answers it seems to me the kid is going essentially by instinct and trying to ‘feel for’ an answer. He/she sees a set of rules that he has to follow but they seem overwhelming and so they either panic or just give up. I feel like further instructions would simply add to that. In my experience I tried to encourage them to ask me questions about the process rather than me asking them and being met by silence and staring. If I could get them to ask questions, stressing that they can be very basic ones, it often allowed me to indentify where they were going wrong and with minor correction they would then be empowered to go forward

      • mpershan

        I feel like further instructions would simply add to that…with minor correction they would then be empowered to go forward.

        You know what’s best for your students, of course.* But I want to explain why I think it’s a good idea to give kids next steps.

        My argument is that this suggested task — verify, extend your answer — is productive no matter whether the kid is working based on feel, instincts, guesses or a solidly coherent theory. It’s just a good thing to do, the kid has a good chance of learning from it.

        As far as the panic goes: that’s something that, if it’s true in my classroom, I need to deal with it. Because I need kids not to shut down when I give suggestions. I’m going to give suggestions to kids on how to improve, because I think that’s crucial for classroom learning, and me/the kids have to figure out how to make that work, emotionally. In other words, this sort of feedback is important enough that I’ll design my classroom around it.

        *Though there is a lot of research on what makes for effective feedback, and a lot of it emphasizes the importance of giving specific, actionable next steps to a student. See my post here

  • Wow. No matter what you say for feedback, this kid is going to get back a quiz without a single problem marked correct, and not much communication to give a positive response to, either. I think you’re right that responding to everything would be overwhelming. So if it were me, I would start with the problem where I can best follow what he was almost certainly thinking and try to respond to that and show him he could move forward from that point, and use comments to point to ideas he’s missing.

    So somewhere on the quiz, I would write something like:

    It’s hard for me to tell for sure what you’re doing on this quiz, because you aren’t writing down steps as you work through problems. If you could work on that for next time, I could give you better feedback and focus in on the parts that are giving you trouble.

    and for #2 I would probably write something like:

    I think on this one, I can guess what you’re doing: probably, you’re taking the 1 and adding it to the 5, and then dividing by 2 to get what x is.

    If so, you’re not far from steps that would work to solve this problem. 2x + 1 > 5 says that if you had 2x and you added 1 to it, you would get a number bigger than 5. So what would that tell you is true about 2x by itself, *before* you added 1 to it? Try writing that statement down here, in either words or symbols….

    _____________________________________________________________________

    Now that you have worked out something that’s true about 2 x’s, what is true about *one* of them (just x)? And is there just one value of x that would make it true (x =….), or could x be any of a bunch of different numbers that all have something in common (x > ___, or x <= ___, etc.)? Because you're quite right that when the problem says "solve the inequality," it's asking you to summarize what must be true about x.

    Hmm. That's way too long, isn't it! Not sure he'd read it, and it sounds more like something I'd say than something I'd write.

    I like your explanation of the verification strategy but would use it for #1 instead. Students can verify an x= type statement for an inequality and find out it's "true" and then stop, without understanding there are other solutions too, if they haven't thought it through. You anticipate that with "find out if other things make it true" but the "testing" is an easier concept to grasp with equalities than inequalities.

    Does your curriculum teach inequalities with border points (or lines in 2-D) where the inequality is true, and how the solutions lie on one side or another (and sometimes also on the border)? That's one way to extend the "testing".

  • Sorry bud, I’m going to go one step back before going forward on this one. If this is a quiz, were there chances of more informal formative assessments prior to the quiz in class in which the teacher could have had a conversation with the student? Something along the lines of noticing the student didn’t show much work or dive deeper into the student (mis)conceptions. I make it a point to try and connect with these students that it’s important to me to know what they’re thinking and without having a conversation with them, showing me their work is the next best thing.

    I wouldn’t be a hardass here. However, my first instinct would be to have a brief conversation with the student about the quiz. I’d pull them aside and say something like, “I noticed you didn’t show me your thinking on this quiz. Will you please pick one question you’d like to explain your thinking on and walk me through how you got your solution?”

    If the student can deliver, then now my role is coaching them on how to better show their thinking on paper. If they picked a question they thought they understood and still demonstrate little understanding, then now my role is pinpointing their misconceptions and maybe solve the question together.

    For me, this quiz feels like a student who just transferred to my class the day of the quiz. I feel that there would have been action taken with this student way before I even handed them this quiz. Which is another thought that just popped into my head. Someone, somewhere, maybe in my credential program said something along the lines, “Only assess students when they’re ready. Not every Friday.”

    I’m more curious about the preventative opportunities that lead up to this quiz.

    Thanks for posting.

  • It’s pretty obvious the kid wasn’t even beginning to try. He was just filling in answers hoping he’d get points. This is not panic. Panic looks different, more writing, effort to “ape” understanding. This is a kid who isn’t trying and doesn’t care.

    To give him feedback on his performance as if this were a genuine effort would be to allow him to get away with his game. I don’t mean you need to be punitive. I’d give him a zero with a note to the effect that he didn’t appear to understand anything, and that I’d be discussing him with this further.

    I’d start to watch him more closely during class, giving him help, during the test, giving him help, noodging him to putting more work into the class.

    • In my experience, panic simply doesn’t always look the same. Panic does, actually, look like this sometimes from students who care a lot — well, until they decide to stop caring, because obviously the teacher thinks they don’t.

      • See, people like you assume that “not trying” is a bad thing, so you judge accordingly and make foolish, snotty, deeply snarky remarks.

        • I am telling you the truth. You may dismiss it as snark, but it doesn’t change the fact that it is true. Students *do* stop caring when they get the message that the teacher assumes they don’t care. Panic does not always manifest itself by trying to “ape” understanding; if you have support for that statement, let’s hear it.
          Of course, some studetns submitting a blank paper may no longer care.
          Can you tell me where I said anything to indicate that I assumed “not trying” was a bad thing? If you have support for that statement, let’s hear it.

  • Well, if you don’t think “not trying” is a problem, then the entire logic of your objection is pointless. Why would it matter to the student if the teacher assumes the student doesn’t care and stopped trying? Clearly because to you, it’s a negative mindset. If instead, a teacher’s “assumption” of the student not caring is value-neutral, just an obstacle to change, then the student wouldn’t “get a message” that would lead to greater disengagement. So your entire logic frame makes no sense.

    My point is simple: a teacher who sees this effort as panic or a misunderstanding of math is going down the wrong road to addressing the problem. Understanding that the student doesn’t care and didn’t try is the first step to changing that, not the last step to dismissal.

    • That’s not what I said; please read more carefully. ” Students *do* stop caring when they get the message that the teacher assumes they don’t care. Panic does not always manifest itself by trying to “ape” understanding; if you have support for that statement, let’s hear it.”

  • I wouldn’t give any specific feedback on this quiz. In fact, I wouldn’t even turn it back to them. I’d say something along the lines of:

    “Tom, how do you think you did on this quiz?”

    “Hmm. I’ll make you a deal. If you redo this quiz, and write down every single step for every single problem, I’ll take it. And then, I’ll review your work until I spot an error. When I spot one, I’ll write down your error. Then, I’ll give it back to you, and you can fix it, and then keep going again. And then I’ll review. We can do this until you get 100%. What do you think?”

    The vast majority of my students have taken this deal. We get some meta-self-analysis: “What was I thinking here?” Some trust in the teacher: “She’ll let me get a 100%!” Some learning, “Oh, so *that’s* how you do it!” Some conversation with the student: “Hey, Mrs. Ostaff, what do you think of this?!”

    I don’t offer this to every student, for every quiz. Just quizzes like this, usually as early in the term as possible. Once or twice is all it takes to vastly improve their work. And, they’re abjectly grateful.

    It doesn’t work for all students, but most of them will try it.

    • I like how focused this routine is on getting the kid to improve their work. You absolutely demand that your students have the chance to improve, and this is so phenomenal.

      You describe the written feedback that you’d give this student:

      And then, I’ll review your work until I spot an error. When I spot one, I’ll write down your error. Then, I’ll give it back to you, and you can fix it, and then keep going again.

      What sort of thinking does a student have to do improve their work through fixing corrections? I bet that we could ask a question that gets this kid thinking about the problem itself in a productive way, instead of trying to patch up the solution.

      • I would argue that you would start the process of thinking about the problem in a productive way when you move it from an impossible task to a teamwork process. Personally, I am a fan of the math journal, and my weekly question is: “What is the most difficult problem this week? How did you solve it? What was the easiest? How did you solve it?”

  • I often tend towards the process- or problem-type related next task like you’ve offered, Michael; helping the kid keep thinking by suggesting that he has the power to check his own work in a specific way a do more thinking.

    In this case, though, having spent a lot of time looking at examples of student mistakes (that show up even without work shown), I think I can make a guess about this kid’s thinking that I’m willing to bet my next move on.

    I think this kid is pretty consistently operating around the variables. I think he is evaluating 2x + 1 as 3 as though x is just a funny label for the 2, and labeling his answer with another x. 2x + 3y evaluates to 5 because again he’s treating x and y as funny labels for numbers you can operate on like normal. So he stumbles to solve systems and write linear expressions because he doesn’t have a meaning for variables beyond labels.

    That’s why I’d hesitate to ask him to try “swapping in 3 for ‘x'” and evaluating that statement — I’m not sure he sees x as a placeholder.

    I can think of 2 possible approaches from here short of pulling the kiddo in for lots of extra help making sense of what variables mean, why we care about what happens to y as x changes, the language of algebra, etc.

    1) Translate the symbolic presentation of one of the inequality problems into words and ask for a logical reasoning solution, to see if the kid has the reasoning chops and is bewildered on the symbols: “I’m going to ask you for a number. When you tell me your number I’m going to double it, then add one. If the result is greater than 5, you win. If not, I win. What are some numbers you can pick so that you will win? What can you say about *all* the winning numbers?”

    2) Ask for the translation of one of the inequality problems, or the equation the kid generated in Q1 into words, starting with “pick any number”, to see if we get any insight into how the kiddo is thinking about the symbols.

    Also, I’m with Andrew on this one — the student is making a pretty consistent kind of mistake. Is there any way this could have been caught earlier?

    • PS to myself: I would probably choose Option 1 of my two options, writing that story on a post-it note with the request that the student 1) answer the question, and 2) indicate which problem on the quiz my “regular language” question is most like and why.

      • I love love Option 1, and your diagnosis is really insightful. Does that diagnosis work for all the problems, though? In Q3 the student seems to be solving 2x+1 = 7, and then on the Q4 he successfully solves an equation (even if it’s the wrong equation). Like with so many student mistakes, this kid’s “misconception” about variables comes and goes. I’d maybe even call it a tendency to ignore the variable and just act on the surrounding numbers, not really a belief. I’m not sure how this impacts the sort of feedback that we’d give. Maybe it means that we want to extend what the kid knows from Q4 about variables and ask him to apply it to Q3 and Q2?