Sunday, November 27, 2011

Homework and more homework.

A year and a half ago, two other experienced teachers and I gave a workshop about teaching geometry. Many of the participants wanted to know how we handled homework. We wanted to have them do geometry activities but decided to allot some time to homework since that was their concern. The other presenters and I decided that the three of us would each get a minute to explain how we managed homework. We did not consult each other beforehand but the three of us were on the same page about virtually everything regarding teaching and learning geoemtry, so we assumed we would support each other on this issue.

It would be hard to find three more different responses than the three we gave. Our descriptions of how we handle homework make the Tea Party and the current administration look like they are in agreement on almost every issue. Each of us had arrived at very different conclusions based on our years of experience.

Having said that, I think I will contribute my thoughts on the subject. I tried a lot of things and all approaches had flaws. I eventually came to several conclusions about homework:
  1. Homework is necessary. Students need time, on their own, to wrestle with mathematical ideas, to put them into perspective, connect them to other ideas and to practice. Some of this can happen in a class, but it is the rare students indeed who is able to make connections and internalize concepts in one class period.
  2. It needs to be done on a regular basis, so students come to class knowing more than they did when they left class the day before. Otherswise there in not much progress happening.
  3. It needs to involve reading and learning from reading. A students who can not read text, study an example, and determine meaning, is not an independent learner and will be forever limited in his or her ablilty to further advance his or her education.
  4. Students need incentives to do homework in mathematics even though they know it is helpful and important. They are human and they are immature. Given the choice, they will usually do something that is more social or more to their own interests than doing the problems that the teacher requires that they do. Even my best students, when homework was not required, spent their time doing the Chem assignment or writing the history paper that was due tomorrow, instead of the math, if they could put it off for another day. "Never put off until tomorrow what you can just as well do the day after tomorrow." --Mark Twain
  5. Homework is a learning experience, not an assessment activity. As such, it should not be graded. It should be evaluated in proportion to the effort that went into it, not in proportion to the number of correct answers.
  6. Students need feedback with regard to the correctness of their work.
  7. My time, both in and out of class, is better spent writing interesting problems, asking good questions, and writing appropriate assessments than it is spent grading homework.
So, after many iterations, I devised a plan that worked tolerably well for me. Here it is:
When students enter the class, they are provided with solutions to the homework that is due. Sometimes this was done by displaying them on the screen in the front of the room. sometimes it was done by placing copies of worked-out solutions in their mailboxes so they could retrieve them as they entered the room. (Every student had a mailbox so that I did not have to waste tome passing out papers.) Were I still teaching, I might resort to putting the solutions on my web page at an appropriate time. At any rate, students always had a chance to see what I thought was an appropriate way to solve the problems. These were more than answers; they were solutions.
I also gavemy students a problem to work on during the first few minutes of class. When class started, I walked around and looked at the homework to see if it looked appropriate. If so, I gave them 2 points. If not, I told them to finish it in a proper manner for tomorrow. If it was turned in the day after it was due, they got 1 point. Otherwise, 0 points.
As I walked around, my students had a chance to ask me about a particular problem or two that they were still confused about. I would make a judgement based on this information about the appropriateness of working a homework problem with the class. Usually we did not.
We were on a quarter system, about nine weeks per quarter. At the end of the quarter, any students who had done all of the homework on time (or at most missed one or had one or two late--that is, 2 points off of the maximum) had earned THE HOMEWORK BONUS! That meant the their lowest quarter test grade would be raised by a grade. I graded by letter grades, not points, so that meant that, for instance, a C became a B.
It worked for me with my students. Most students did their homework daily and were very concerned about not losing that bonus.
Oh, there was a dark side. If a student missed more than a week's work, five assignments, their highest test grade was lowered by a full grade for every one they missed after five.
I had wanted the homework score to be entirely positive: If you do this, it will advance your work by a bit. I found that after students lost the bonus, some decided it was no longer necessary to do any homework, so I had to include the negative part. It worked as well. Students do not like losing something they have already earned.
I am not sure how much of the homework I looked at had been copied. I never did find a good way to combat that problem, other than giving two or three unannounced quizzes every week. All of the quizzes counted as one test. They were short and I thought of them as formative more than summative, but they did count. And I did learn quite a bit about what my students had learned and what they had not yet learned, based on these quizzes.
I think this system worked very well for me. I hope there is a part of this you can use as well.

Friday, November 11, 2011

Of Babies and Bathwater

Thanks, John, for those kind words.  And yes, I plan on presenting more proofs.

These last two weeks have been busy with end-of-quarter grades, projects, etc.  (A forthcoming post will be "High-school level projects that involve actual mathematics," but I digress.)  This post is reflecting on the not-entirely-successful first iteration of our "no homework grade" policy: nightly homework assignments don't count towards students' grades, but frequent unannounced quizzes use representative homework problems as an incentive to complete assignments and an assessment of whether students know how to do the math.

The data are in and the following seem clear:
  • Students are completing less homework.  How much less is unclear, because in the past, the marks on students' papers didn't always correspond to thoughtful effort expended on mathematical problems, but in my upper-level classes, it's typical for only about half the students to have attempted a significant number of problems, and in previous years, it was more like 80%.  In the past, I doubt that the typical homework assignment was copied/scribbled from answers by 30% of my students.  Lower-level classes are doing better on most days.
  • Students perceive the policy as "you don't have to do homework," which seems like a misreading to me.  More accurate would be "homework isn't graded and factored into your overall grade." 
  • Students are doing less well on in-class tests than they did on last year's tests, although results vary by class.  Classes that are giving a lot of homework quizzes are finding less dropoff, but those same classes have younger students.
In his fabulous book, So Much Reform, So Little Change, Charles Payne dissects the etiology of failed reform efforts, discovering that a pervasive symptom is simply discarding changes that don't work, rather than examining and adjusting them.  In that light, our department is trying to revise our new homework strategy rather than revert to one we found problematic.  Some questions and possible answers:

1.  Why don't students do homework under the new policy, when they can see their grades dropping?  First, they may not see the connection between doing homework thoughtfully and actually getting better at math: it's telling that large numbers of students describe the policy (to parents, counselors, and teachers) as "you don't have to do homework" rather than "homework is important to learning, but you're only graded on what you learn, not what homework you do."  Second, they have a lot of other work--we're suffering from being the "first movers" in responding to Race to Nowhere.  When a student is up at midnight and choosing whether to do math or go to bed, the threat of a possible homework quiz is clearly not enough incentive to do a handful of math problems.

2.  What can we do to improve the situation without throwing everything out?  First of all, we can communicate better about what we think it takes to learn serious mathematics.  Maybe we should take a cue from the advertising folks, and make posters saying "192 minutes is not enough" or comparing time spent doing math to time spent doing other valuable activities?  What if we change the homework quiz policy so that homework quizzes are frequent but expected, for example every Monday and Friday?  What if we specifically identify which homework problem serves as the basis for each quiz question?

3.  What other ways are there to get students to do math outside of class without increasing incentives to cheat or skate by?  Webassign?

This is a tough time for us:  we want to do things differently and better, so we need to figure out how to adjust our course rather than simply u-turning.  Any ideas and suggestions are welcome; we'll take them in the comments.