More than you'll ever need - Math Education Edition

It is impossible (or at least frivolous) to think about teaching without also thinking about learning.  Since I am currently learning to play the harmonica, I think of the harp anytime I'm thinking about learning.  Recently I was watching an Adam Gussow video about the ever elusive blue third, and he said something that gave me the answer to that age-old question, "How much math do I need to know so that I can _______?"

It is a question that comes up regularly in my upper division math courses.  Most of our upper division math students at Peru State College are aspiring high school or junior high math teachers.  At some point in vector spaces, non-Euclidean geometry, or two variable linear systems of differential equations, someone will say something like, "All I wanna do is teach algebra in high school.  Why do I need to know this stuff?"  For the answer, enter Adam from Satan and Adam.

Realistically, you won't get much out of the video if you don't have any knowledge of playing the harp or at least some basic knowledge of music theory.  Kind of like an algebra lecture, I suppose.  So, I will summarize for you, my gentle readers. Though I sometimes think it's more like gentle reader.  As in singular.  Thanks, Mom!

What is a blue third?

Scales are easily one of the most important building blocks in music.  In nearly all music that isn't based on power chords, much of the character in a harmony or a chord comes from the third note in the scale.  As an aside, I've read that Pete Townshend was greatly influenced by Henry Purcell's use of fifth chords.

Arguably the most common scale is the major scale.  In the key of A, the major scale is A-B-C#-D-E-F#-G#-A or do-re-mi-fa-so-la-ti-do.  Major scales tend to be used in happier songs.  A natural minor key (different from the harmonica minor) has the 3rd, 6th, and 7th notes lowered one-half step.  So, the natural minor scale in A is A-B-C-D-E-F-G-A.

The punch-line to all of this is that the third note in both scales is either C# or C natural.  But in the blues scale, the third note is neither C# nor C natural.  The so-called blue third is a note in between these two notes.   This makes instruments like harmonica and guitar particularly suited to the blues since a piano (for example) is not designed to play a blue third.  To sound a blue third on a harmonica in second position (cross harp), a player must bend the three-hole draw down about a quarter step.

Power in reserve

Here's the rub.  Between four and five minutes into this video, Adam provides the answer I now have for anyone who asks, "Why do I need to learn all this?" Because you need to have what Adam calls power in reserve.

Have you ever wondered why a family station wagon has a speedometer that goes all the way up to 160 mph?  Does anyone need to go that fast in a country where most states have a speed limit of 70 or 75 mph?  Of course not.  But what would happen if the engine in your family wagon were designed for a top speed of 80 mph?  The engine would fall apart after a few weeks, because you can't run an engine near top speed all the time.

Let's get back to the harmonica.  If I can bend the three-hole draw down one-quarter step, then I can play the blue third.  So, I'm done.  Why learn how to bend it any farther?  Well, if I want to apply vibrato, I need to pull it down a little more and shake.  To take advantage of the vocal qualities of the harp, I can play the minor third and release the note up to the blue third.  Not my point-- Adam mentions this in his videos.

My point is that the quarter step bend on three-hole draw is not the end of learning the blue third.  It's the beginning.  Just like second semester calculus is not the end of content knowledge for 7-12 mathematics education.

Learning requires growth - Yoda knows it

In math, as well as music, history, or rhetoric for that matter, one mark of a well-trained person is that he or she knows more than he or she will generally need. For an aspiring high school math teacher, teaching constantly at the upper limit of one's knowledge is a recipe for frustration at best, disaster at worst.

As Maslow might have said, you can choose between safety and growth.  Stepping out of our comfort level at least on occasion is necessary for us to become the type of people we aspire to be.

You might not be surprised to hear that one of my all-time favorite movies is The Empire Strikes Back.  My two favorite scenes from all of Star Wars happen in the middle of The Empire Strikes Back when our hero, Luke Skywalker, meets the inimitable man among muppets, Yoda.

In the first of these scenes, Yoda tests young Skywalker's patience by acting the impish prankster.  "Awww.  Cannot get your ship out," has long been one of my favorite catch-phrases.  The second of those scenes is where Luke discovers that the imp is, indeed, the great Jedi master, Yoda.  In the ensuing conversation, Luke and the disembodied spirit of Obi-Wan Kenobi attempt to convince Yoda to train Luke in the ways of the Jedi.

No one questions that young Luke has the aptitude to succeed.  He is, after all, the son of Anakin Skywalker.  However, Yoda is uncertain that Luke will finish what he begins.  When he voices this concern, Luke responds, "I won't fail you.  I'm not afraid."  To which Yoda plainly states, "You will be.  You will be."

Yoda realizes, as all good teachers, trainers, and coaches realize, that the true mettle of a student is not evident until the learning becomes difficult.  And learning is always difficult at times.  At those times, the single worst thing you can tell your students is that they don't really need to learn that anyway.  I'll be blogging in more detail on this subject in a later entry.  For now, I'll note that a student dropping out of school is not as dramatic as losing an apprentice to the dark side of the force, but it is a tragedy nonetheless.

The simple point is this.  If you want to be a teacher, you need to enable your students to grow.  As a matter of teaching philosophy, mathematics to me is like faith to religious people.  You can't know too much about it.  Our nation will not come out of recession, develop sustainable energy technologies, and lead the world in the 21st century with a gaggle of math teachers who don't need to know more than second semester calculus.

Let's do this! - Creating a community of practice

One of the goals that all educators should aspire to is the creation of a community of practice among their students.  It is, in my opinion, the ultimate expression of the active, collaborative, and inquiry-based learning strategies.

What is a community of practice?

In brief, community of practice is a phrase coined by anthropologists Jean Lave and Etienne Wenger.  There are three requirements for a community of practice. The first requirement is commitment to a domain.  For example, computer programmers would share the common interest of programming.

The second requirement is that the members must actually be a community.  That means they interact in a way that allows them to share information, request help, etc.  Simply having an interest in professional wrestling does not make you part of the community of pro wrestling fans if you never interact with anyone else in the community.

The third requirement is that of practice.  The members of the community must be practitioners of their domain.  So, fans of the Chicago Cubs can be a community, but they are not a community of practice, since the domain is not a practice.

Communities of practice in the classroom

So, what does a community of practice look like in the classroom?  About what you would expect, really.  Students who work together on their studies are a community of practice.  The subject matter at hand is the domain, so all that remains is to build a community among students.

Often, students will form learning communities without any external influence.  In particular, students who have been in college for a while tend to forgo the lone wolf approach that tends to work in high school.  At Peru State College, our science majors in particular seem eager to work together in their studies

Even when there is no need for an instructor to induce collaboration, it may be necessary for other purposes.  For instance, accreditation entities tend to expect documentation of such activities.  They are not likely to accept your word for it when you tell them that your students work together on their studies.

What can we do to create a community of practice?

So, how do I go about creating a community of practice, um, in practice?  My strategy for establishing a spirit of collaboration among students is to borrow from Maria Andersen's approach to board work.  In this activity, students work on problems in pairs at the board.  Half of the students (one in each pair) are designated to remain stationary while the others move around the room, periodically changing partners.

Working at the board allows everyone in the room (especially the instructor) to see what the students are up to.  The instructor can keep track of student progress, and other students can look outside of their groups for help if they get stuck.  You also don't have to worry about any dysfunctional partnerships since no pair of students stays together for too long.  In fact, Muskegon Community College has gone so far as to design some classrooms with this approach in mind.

From a formal perspective, it makes sense to have students work on projects in groups.  This allows them to get more practice in working on teams as well as enabling them to tackle more involved assignments.

It is worth noting that students in online courses can also benefit from forming a community of practice with their peers.  The lack of face-to-face interaction does not preclude interaction among students, but it does require a more tech savvy approach.

We have the technology

One thing that all modern mathematics classrooms must account for is the availability of powerful and widely usable technologies for solving mathematics problems.  Consider that anyone with internet access can use Wolfram Alpha not only to solve equations, but also to provide solution steps.

Do technologies such as this make mathematical training obsolete?  Not hardly.  In fact, I would argue that such technology makes mathematical training even more important than it has ever been.  As Maria Andersen puts it, "If you can be replaced by a computer, you're likely to be replaced by a computer."

Rather than fear the technology or simply pretend it doesn't exist, I think the best plan is to embrace it.  Of course, students still need to know the fundamentals of solving algebraic equations.  But technology can both save them time as well as help them learn.

How to bring it into the classroom

And now we get the practicalities of the situation.  What am I going to use in the classroom, and how will I incorporate it into the student learning experience?  For various reasons, I've settled on Wolfram Alpha.  The primary motivations are that W|A is inexpensive (no additional cost if you have access to the internet) and fairly easy to use given the sophisticated input interpretation.  There are also some side benefits such as being able to generate graphs from real data on the fly.

In the past, I have primarily used W|A for classroom demonstrations and generating graphs quickly.  It works well, the only obstacle being that many of the classrooms I teach in don't have computers and projectors already set up.  While I encourage students to experiment with the software, I don't devote significant class time to working with the software.  This is primarily due to the lack of a regularly available computer lab.  That and the fact that there is no guarantee that students will use the software unless I force them, which I'd rather not get into.

Prototype demonstration

As an example of what I will do with W|A, consider the topic of graph transformations.  A function plotting utility is the perfect means of quickly showing the effect of a transformation on a graph.  For example, consider the following.

As we see, W|A color codes each plot.  This makes graphs of multiple functions much easier to digest.  And the input is simply the functions separated by commas.  x^2, (x-3)^2, (x-2)^2+5, 3(x-2)^2+2  No confusing syntax involved.

Where to go from here

This particular aspect of planning for College Algebra is no different from my plans in recent semesters.  I hope sometime to have ready access to a computer lab so that students can experiment with the program in class without pulling out their internet capable cell phones.  The good news is that even if that doesn't happen, the technology is still a great help to my teaching.  And the few students who do use it in and outside of class seem to get something positive out of it.

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.

Why are we here? - Planning Day 1

It is one of life's great questions

Today's post is about what I feel is the most important question to discuss with students on the first day of any course.  That question is "Why are we here?"  Not here on this Earth, but here in this room.  More directly, why are my students supposed to learn College Algebra-- or Differential Equations or Cultural Anthropology for that matter.  In the ever crowded and ever shrinking modern curriculum, why are three precious credit hours squirreled away for the material you will be studying (or not) for the next 15 weeks?

In my opinion, we owe it to our students at the least to let them know why we feel the material in our courses is relevant to them.  And "I had to learn it when I was your age by cracky, and you're gonna learn it too." is the wrong reason.  From most modern perspectives on learning, it makes sense for students to be active participants in this discussion.  Ironically, it is difficult to have a discussion of much substance without cutting into the amount of material we can present, so I intend to keep this part of Day 1 brief.  About a half hour sounds right, and that's a good way to introduce the syllabus as well.

What this means for Math

As numerically literate people in general and mathematicians in particular, we should all have a short answer to the question "Why is Math important?"  And yet, it's akin to asking musicians why they study Bach instead of Michael Bolton.  It's as if we can't get over the novelty of a genuine question whose answer seems so obvious to us.

Math is important for many reasons both philosophical and practical.  For starters, knowledge of math is necessary to balance a checkbook, build a barn, and estimate the time to drive from Louisville, KY to Auburn, NE.  Incidentally, it's about 12 hours.  On top of that, there is a large and old body of evidence suggesting that a free and Democratic society depends on a populace that is numerically sophisticated enough to understand that $0.002 is not the same thing as 0.002 cents.

But what about your subject in specific?  For me, College Algebra is necessary for many things, both on its own merits as well as a gateway to things like statistics, medicine, and finance.

Back to Day 1

So, how do students get the message quickly, on the first day of class?  I suggest picking an example that is at least partially relevant to them.  For instance, how far is it from home plate to second base on a major league baseball diamond?

A good brain teaser might also work.  Suppose two players each flip a coin and show the other player the result.  If both people must simultaneously guess the result of their own flip, is there a strategy that will guarantee that at least one of them is right all the time?

What about something more like a construction?  The handshake lemma is non-trivial, but most students can derive it in a group in about 15 minutes or so.

Another possibility is either demonstrating or showing a video demonstration of math on a current event topic. The Mathematics of War TED talk of Sean Gourley can be a crowd pleaser.

When the student in prepared...

When the student is prepared, something good will probably happen.  For myself, the goal of Day 1 in any course is to let students know what they're in for and let them know that they will have to work if they want to succeed.  But perhaps more importantly, my job is to let students know that the knowledge and skills I will present to them are worth working for.

Exactly how I do that remains to be seen, but at least I'll have something to think about for next time.  In between, I've got to decide what I'm going to do about homework.