Monday, July 11, 2011

Here's where the fun begins - Homework

One of my historical issues in preparing to teach any course is in the assigning of homework.  Perhaps more than anything else, the amount of effort students are willing to put into well-planned practice determines their success in math courses. By well-planned, I mean thinking about and working on the right kinds of problems that will help them prepare for class, succeed on exams, etc.

Why is homework important?

The easiest way to get this point across is a little demonstration I picked up at a college algebra workshop in Wisconsin two years ago.  First, watch this short video.


What you have just seen is a demonstration of how to play Love Me Do on the harmonica (10-hole diatonic in C major, specifically).  This demonstration was accompanied by a detailed description with visual aid of what Nick is playing.  Just like an algebra lecture, isn't it?

If you are like many students, you just watched the video.  It all made sense, so that means you've got it, right?  All you need now is $5 and a trip to the local music store to get your own harp and you can play just like John Lennon.

What would actually happen

After watching this video, a raw beginner would pick up a C harmonica and make a decent stab at playing Love Me Do.  If you had never played a harp before, you would have trouble hitting single notes.  Most of your playing would be poorly articulated chords since playing clear single notes is a skill that takes a good week or two of practice for a brand new player.  You would have to master one of the three different methods of sounding good single notes, and one of those-- called U-blocking-- is only possible if you are capable of rolling your tongue.  The ability to roll your tongue is genetic and can't be learned.

You also might need to listen to the recording a few times if the song isn't familiar to you.  Things like rhythm and subtlety are hard to pick up from a 3 minute lesson.  Also, there are a few parts of the song that Nick has not shown you how to play.  If you want to play along with the record, you would either need to look them up or discover them through trial and error.

And the best part is that once you acquired all these skills, Love Me Do still wouldn't sound quite right.  You see, professional quality harmonicas are more expensive than $5 for a good reason.  Cheap harps are notorious for being hard to play due to air leakage, poor reed gap tolerances, etc.  They also taste terrible. To sound like John Lennon, you would need to drop at least $15-$20.  In the video lesson, Nick is using a Hohner Golden Melody that runs about $36.


Realistically, none of this likely surprises any of you who have tried to pick up an instrument.  Just like mathematics, learning an instrument requires both solid grounding in basic skills as well as significant practice time.  And yet, many of our students expect the mathematical equivalent of impersonating James Cotton after less than 90 minutes of instruction and no practice.

So what do I do about homework?

With my philosophical musings out of the way, let's get to the practical matter. What am I going to do with homework in my course?  There are a few guiding principles I have in designing a homework scheme.  I'll discuss each in brief.

1.  Sizable problem sets with each section
2.  Not grading homework assignments
3.  Basing about 85-90 percent of exams directly on homework problems

Sizable problem sets

Large problem sets are intended to provide guided practice to the student.  A student who successfully completes all problems from each section should have little trouble earning at least a B on each exam (see principle 3).  I include weekly quizzes as part of my course in order to prep students for exams, give them the opportunity to earn a few points in a low stakes environment, and keep track of attendance.  In theory, weekly quizzes incent students to complete homework in preparation.

More importantly, I want homework sets to have enough of each type of problem that a student struggling with a particular concept has an obvious place to go to find more problems representative of that concept.  I also like to include a mix of simpler and more sophisticated problems of each type.  Students who begin with more preparation can work the more difficult problems first and move to the simpler problems if some concepts need refreshing.

Not grading homework

Realistically, this is almost exclusively a time saver.  If I had an army of grad students at my disposal, I wouldn't think twice of assigning problems to turn in for a grade.  Also, anecdotal evidence suggests that students are all too willing to copy someone else's homework to turn in along with all manner of other methods of avoiding completing their homework.  Actually, if I had an army of anything at my disposal, I hope I could come up with something more interesting to do with them...

I like to think that not grading homework assignments frees me to include more robust and diverse problems for students to guide their studying.  Also, students can work on a problem until they understand it without putting in still more time to prepare it to turn in for a grade.  This is an important skill, but that's what the quizzes are for.

Exams based on homework sets

Basing exams on homework is a necessity in course design.  For starters, I want to assess student mastery of certain concepts, so my lectures, homework problems, and exams should all be designed with that goal in mind.  Also, students should not have to guess what's important for them to learn.  In my classes, I want students who master the homework problems to be able to say that they have learned what I intended.  And exams are the evidence that this has occurred.

Finally, I think it's important that a student who earns an A on my exam has done more than master homework problems.  The point of learning is not to pass an exam, but to be ready to put knowledge into practice.  And no one (outside of an algebra teacher, of course) ever gets a job where the boss wants you to factor polynomials.  A boss who wants you to think of a function to predict sales decay after the end of an advertising campaign might actually happen.  And that's why at least 10 percent of my exams don't come directly from the homework.

So what am I gonna do?

I sense that students have not been working on their homework problems.  Of course, I have nothing but my intuition and their exam scores to go on since I haven't been collecting and grading homework problems.  My current plan is to adopt a middle of the road strategy that takes advantage of my current practices.

Since I already include a weekly exam into my courses, I'm simply going to make each quiz a sample of problems from the homework.  This way, students who complete the assignments will get an immediate, tangible benefit from doing so.  In an ideal situation, students who have fallen behind in homework will see their peers' better quiz grades and rethink a nonchalant approach to homework.

If nothing else, maybe someone will learn to play harmonica.

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