## Rubrics & skill standards – a rollercoaster case study.

• I gave a quiz not long ago with the following question adapted from the homework:

The value of 5 points for the problem came from the following rubric I had in my head while grading it:

• +1 point for a correct free body diagram
• +1 for writing the sum of forces in the y-direction and setting it equal to may
• +2 for recognizing that gravity was the only force acting at the minimum speed
• +1 for the correct final answer with units

Since learning to grade Regents exams back in New York, I have always needed to have some sort of rubric like this to grade anything. Taking off  random quantities of points without being able to consistently justify a reason for a 1 vs. 2 point deduction just doesn’t seem fair or helpful in the long run for students trying to learn how to solve problems.

As I move ever more closely toward implementing a standards based grading system, using a clearly defined rubric in this way makes even more sense since, ideally, questions like this allow me to test student progress relative to standards. Each check-mark on this rubric is really a binary statement about a student relative to the following standards related questions:

• Does the student know how to properly draw a free body diagram for a given problem?
• Can a student properly apply Newton’s 2nd law algebraically to solve for unknown quantities?
• Can a student recognize conditions for minimum or maximum speeds for an object traveling in a circle?
• Does a student provide answers to the question that are numerically consistent with the rest of the problem and including units?

It makes it easy to have the conversation with the student about what he/she does or does not understand about a problem. It becomes less of a conversation about ‘not getting the problem’ and more about not knowing how to draw a free body diagram in a particular situation.

The other thing I realize about doing things this way is that it changes the actual process of students taking quizzes when they are able to retake. Normally during a quiz, I answer no questions at all – it is supposed to be time for a student to answer a question completely on their own to give them a test-like situation. In the context of a formative assessment situation though, I can see how this philosophy can change. Today I had a student that had done the first two parts correctly but was stuck.

Him: I don’t know how to find the normal force. There’s not enough information.

Me: All the information you need is on the paper. [Clearly this was before I flip-flopped a bit.]

Him: I can’t figure it out.

I decided, with this rubric in my head, that if I was really using this question to assess the student on these five things, that I could give the student what was missing, and still assess on the remaining 3 points. After telling the student about the normal force being zero, the student proceeded to finish the rest of the problem correctly. The student therefore received a score of 3/5 on this question. That seems to be a good representation about what the student knew in this particular case.

Why this seems slippery and slopey:

• In the long term, he doesn’t get this sort of help. On a real test in college, he isn’t getting this help. Am I hurting him in the long run by doing this now?
• Other students don’t need this help. To what extent am I lowering my standards by giving him information that others don’t need to ask for?
• I always talk about the real problem of students not truly seeing material on their own until the test. This is why there are so many students that say they get it during homework, but not during the test – in reality, these students usually have friends, the teacher, example problems, recently going over the concept in class on their side in the case of ‘getting it’ when they worked on homework.

Why this seems warm and fuzzy, and most importantly, a good idea in the battle to helping students learn:

• Since the quizzes are formative assessments anyway, it’s a chance to see where he needs help. This quiz question gave me that information and I know what sort of thing we need to go over. He doesn’t need help with FBDs. He needs help knowing what happens in situations where an object is on the verge of leaving uniform circular motion. This is not a summative assessment, and there is still time for him to learn how to do problems like this on his own.
• This is a perfect example of how a student can learn from his/her mistakes.  It’s also a perfect example of how targeted feedback helps a student improve.
• For a student stressed about assessments anyway (as many tend to be) this is an example of how we might work to change that view. Assessments can be additional sources of feedback if they are carefully and deliberately designed. If we are to ever change attitudes about getting points, showing students how assessments are designed to help them learn instead of being a one-shot deal is a really important part of this process.

To be clear, my students are given one-shot tests at the end of units. It’s how I test retention and the ability to apply the individual skills when everything is on the table, which I think is a distinctly different animal than the small scale skills quizzes I give and that students can retake. I think those are important because I want students to be able to both apply the skills I give them and decide which skills are necessary for solving a particular problem.

That said, it seems like a move in the right direction to have tried this today. It is yet one more way to start a conversation with students to help them understand rather than to get them points. The more I think about it, the more I feel that this is how learning feels when you are an adult. You try things, get feedback and refine your understanding of the problem, and then use that information to improve. There’s no reason learning has to be different for our students.

## Having conversations about and through homework

I’ve been collecting homework and checking individual problems this year. I grade it on completion, though if students tell me directly that they had trouble with a question before class (and it is obvious it isn’t a case of not being able to do ANY of it because they waited until the last minute to try) I don’t mind if they leave some things blank. I did this in the beginning since I had heard there were students that tried to skip out on doing homework if it wasn’t checked. We do occasionally go over assigned problems during class, but I tend not to unless students are really perplexed by something.

I have lots of opinions on homework and its value. Some can use the extra practice and review of ideas developed in class. Some need to use homework time to make the material their own. In some cases, it gives students a chance to develop a skill, but in those cases I insist that students have a reliable resource nearby that they know how to use (textbook, Wolfram Alpha, Geogebra) to check their work. I don’t think it is necessary to assign it just to “build character” or discipline. I read Alfie Kohn’s The Homework Myth, and while I did find myself disagreeing with some aspects of his arguments, it did make me think about why I assign it and what it is really good for. I do not assign busy work, nor do I assign 1 – 89 – each problem I assign is deliberately chosen.
Among the many ways I try to assess my students, I admit that homework doesn’t actually tell me that much about the skill level of a student. Why do I do it then?

My reason for assessing homework is for one selfish reason, and I make no secret of it with my students:

### The more work I see from students relating to a concept, the better I get at developing that concept with students.

I would love to say that I know every mistake students are going to make. I know many of them. If I can proactively create activities that catch these misconceptions before they even start (and even better, get students talking about them) then the richness of our work together increases astronomically. You might ask why I can’t get this during conversation or circulation with students during the class period. I always do get some insight this way. The difference is that I can have a conversation with the student at that point about their thinking because he or she is in the room with me. I can push them in the right direction in that situation if the understanding is off. The key is that most of my students are alone when they do their work, or at least, have only online contact with their classmates. In that situation, I can really see what students do when they are faced with a written challenge. The more I see this work, the better I get.

I am not worried about students copying – if they do it, it always sticks out like a sore thumb. Maybe they just aren’t good at copying. Either way, I don’t have any cases of students that say ‘I could do it in the homework, but can’t do it when it comes to quizzes or tests.’ Since I can see clearly when the students can/can’t do it in the homework, I can immediately address the issue during the next class.

The other thing I have started doing is changing the type of feedback I give students on homework. I still fall into the habit of marking things that are wrong with an ‘x’ when I am not careful. I now try to make all feedback a question or statement, as if I am starting a conversation with a student about their work through my comments, whether positive or negative:

• Great explanation using definition here.
• Does x = 7 check in the original equation? (This rather than marking an x when a solution is clearly wrong.)
• (pointing out two correct steps and then third with an error) – mistake is in here somewhere.
• You can call “angle CPK”  “angle P” here.
• Good use of quotient rule – can you use power rule and get the same answer?

The students that get papers back with ink on them don’t necessarily have wrong answers – they just have more I can chat with them about on paper. The more I can get the students to understand that the homework is NOT about being right or wrong, but about the quality of their mathematical thinking, I think we are all better off.

This does take time, but it is so valuable to me, and I think the students not only benefit from the feedback, but appreciate the effort on my part. I don’t check every problem, just key ones that I know might cause trouble. If a student has everything right on the questions I am checking, it’s a chance to give feedback on one of the others. If there’s nothing to say because the paper is perfect (which is rare), I can praise the student for both their clear written solutions, hard work, and attention to detail.

I decided at the beginning of this year to look at more student work, and checking homework in this way is letting me do this. I am lucky to have prep time in the morning, and I have committed to using morning time for looking at student work almost exclusively. I have had to force myself to do this on many mornings because it’s so easy to use the time for other things. Some of my best ideas and modifications to lessons come after seeing ten students make the same mistake – it feels good to custom fit my lessons to the group of students I have in front of me.

In the end, it’s just one more way the students benefit from having a real teacher working with them instead of a computer. Every mark I make on the paper is another chance to connect with my students and conversation that can help make them better thinkers and learners. I don’t think I really need to justify my presence in the classroom, but it feels good to say that this is one of the reasons it’s good I’m there.