I have a short update today. Enjoy the relief from my usual cavalcade of verbiage
Thanks to our Chief Academic Officer, I just saw the article College-Wide Grading Standards on the Foundation for Critical Thinking website. The article described almost exactly what I was planning to post for part 2 of my series on assigning grades. Instead of posting part 2 of my series, I will modify my entry to incorporate this excellent resource.
The article provides the qualitative descriptions of grades A through F, so my current plan is to discuss the differences in each grade category. I hope to report on my results of developing an operational grading strategy based on these standards.
In other news, I've decided that my Linear Algebra course next semester will include a requirement that students create a blog of their learning progress in lieu of turning in homework problems. I will still provide homework problems for the students to work on, but they will blog their progress instead of turn their solutions in for a grade.
All that and much more is forthcoming, so don't touch that dial.
Sunday, July 31, 2011
Friday, July 29, 2011
Grading with flair - Part 1 - 15 is the minimum
Today's blog entry is the first in a series of at least two where I will discuss my philosophy of evaluating student work and the assigning of grades. As Tolkien would say, it is a tale that grew in the telling. Thus, I've decided to allow myself multiple writings as I refine the Sinkhorn Rubric.
One of my all-time favorite quips goes a little something like "If you think your teachers are tough, wait until you get a boss." Too true. You see, all good teachers need to be patient with learners. As I mentioned in my last post, Yoda was patient with Luke Skywalker. Purposefully, I did not go into great detail as to why great teachers are patient with their students. My purpose was that I did not wish to steal any thunder from this blog entry.
The link here between Yoda and evaluation of students is that honest, constructive assessment of student work requires many things: among them are clear expectations, acceptance of creativity, and patience. And the most important of these is patience. Students, like children and employees, benefit from high expectations. But it is counter-productive to insist that an instructor's evaluation of a student does not include at least a certain level of opinion.
In this first post, I discuss the biggest concern I have with evaluating student performance. I will beg your indulgence, since I take a longer time than usual to get to the point. So please, be patient.
But first, we need to talk about your flair
If you have never seen Mike Judge's feature length directorial debut, you should stop reading this and watch Office Space as soon as safety and decency allow. In the movie, Jennifer Aniston's character, Joanna, is a waitress working in one of those restaurant chains. You've been there. It's the place where members of the waitstaff are evaluated solely on their willingness to behave in the most obnoxious and ingratiatingly sycophantic manner allowed by law.
The emblem of this behavior is flair. In the Mike Judge world of a Chotchkies restaurant (Judge himself plays Joanna's supervisor), servers are required to wear at least 15 employee-supplied buttons-- the so-called flair-- on their uniform suspenders. "People can get a burger anywhere. They come to Chotchkies for the atmosphere." Suspender flair is presumably a big part of this atmosphere.
Joanna thinks, correctly, that the "atmosphere" of Chotchkies both debases their employees and insults the intelligence of their customers. Stan, the supervisor character, picks the wrong day to criticize Joanna for wearing only the minimum 15 pieces of flair, and her response is honest, priceless, and a textbook example of a person railing against a divisive and poorly designed system of assessment.
Suffice it to say, Stan is not a born leader. Mike Judge's character is a person who apparently got ahead by scoring high on the poorly designed Chotchkies rubric of evaluating employees. The Chotchkies rubric does not identify and reward leadership, since leadership is apparently not something valued in the mid-level management at this particular establishment.
Guerrilla Grading
In our accreditation-driven education system, exhaustively detailed grading rubrics have become a holy grail of sorts. In my opinion, it is extremely important that grading be both as fair and objective as possible. Thus, a rubric is a natural means of establishing a standard to which students can compare their work as a part of their own self-assessment.
This is a good thing for students and instructors. But I like to remember that professional educators are, well, professionals. And one of the most important things that all professionals share is the need to operate with some degree of autonomy. Simply put, educators earn a living at least in part because they know a good paper when they see one.
For me, the difficulty in grading student work is not determining right answers from wrong answers-- or even flawless logic from an argument that needs to be refined. My difficulty is in assigning points. More properly, my difficulty is in deciding how many points to subtract for a specific error or omission. Is a sign error a one point mistake or a three point mistake? The answer to that depends on the problem. Suppose I ask for the roots of x^2+4, a sum of perfect squares.
This is what I'm looking for.
If I instead get the following, the difference is merely a sign error. But it is exactly what I don't want.
A question of this nature is specifically designed to test if a student can recognize that a certain quadratic equation has no real roots. And I want the student to do this without having been provided the graph. Students can discover this algebraically by solving the equation, visually by plotting the graph themselves, or intuitively by recognizing that x^2+4 is a vertical translation of x^2.
The Sinkhorn Rubric[TM] - Version 1.0
As it turns out, life is a pass/fail course. According to Jorge Cham, this is also a reasonable characterization of life in graduate school.
The Sinkhorn Rubric is based on the pass/fail system. A Jorge Cham pass/fail system is kind of like getting either a C or an F. For me, an A is good, a C is good enough, and an F is not good enough. The first and most basic version of the Sinkhorn Rubric is as follows.
Beer, exams, and beauty contests
In a previous life, I was a blue-ribbon homebrewer. My summer wheat took first in the specialty category at the Kentucky State Fair many moons ago. I also was a beer judge as well. Judging homebrew is kind of like reviewing wine for a magazine, but homebrewers will drink pretty much anything that is both non-toxic and made with malted barley.
In the Homebrew division of the Kentucky State Fair, there are three rounds. In round one, each beer is judged against a style recognized by the American Homebrewers Association (AHA). Points are awarded based on how well the entry adheres to the style guidelines in categories such as flavor, bouquet, and appearance. The X number of beers with the X highest scores move on to round two.
Round two is the medal (ribbon, actually) round. Judges in each category decide on their three favorite beers, then arrange the top three in order of first, second, and third. In round three, the blue ribbon winners in each category compete for the coveted Best in Show award.
You see, round one is like grading exams. You have to decide what to take points off for, and you also decide how many points to take off for each mistake. Even though the AHA puts out every effort to make the process as objective as possible, it is still highly subjective. In fact, all judges in each category of round one are required to make certain that their scores fall within a certain tolerance of each other. And just like grading exams, judging beer for long periods of time has a tendency to dull the senses. For some strange reason...
Rounds two and three are like judging a beauty contest. In round two, all judges in, for example, the light ale category get together and decide which of the beers in their category are the three best. These are the ribbon winners. Then the judges place the ribbon winners in order of preference-- first, second, and third place.
Now let me ask you, my gentle readers. Would you rather judge round one or round two? That is, would you rather grade papers or judge a beauty contest?
AC/DC and fully ordered sets
From a mathematical perspective, the major difficulty associated with schemes for scoring homebrew entries is that the scores are generally whole numbers. As a subset of the real numbers, whole numbers are what is called a fully ordered set. That means that whenever two numbers in a fully ordered set are compared, either the numbers are equal or one is larger than the other. So, whenever two distinct (i.e. non-equal) whole numbers are considered, one of them must be larger than the other.
The simplicity of a fully ordered set is quite beautiful to a decision maker. For example, consider that you wish to buy a 2010 Buick LaCrosse. If Buick A costs $19,000 and Buick B costs $19,500, then Buick A is less expensive than Buick B. You will never need an expert to tell you that a $19,500 car is more costly than a $19,000 car.
And that's the beauty of a fully ordered set. But as soon as there is a matter of opinion, things get a little more dicey. We can easily look up the numbers to see that Australian rock band AC/DC's 1981 album For Those About to Rock reached a higher maximum position on the Billboard album charts than Back in Black. For Those About to Rock reached number 1. Back in Black did not reach number 1 despite becoming the second best-selling album in history.
But if you sit down and talk to any serious (or even a casual) AC/DC fan, you will be in for a long discussion of which of these is the better album. An album grading rubric might help, especially if all manner of AC/DC experts were involved in the creation of this rubric. But the determination of which is the better album ultimately must be, by nature, a matter of opinion. That's why music departments teach Music Appreciation instead of Music Scoring and Ranking.
Partially ordered sets and a cheap, used Buick
When the rubber hits the road, any kind of sophisticated decision comes down to a matter of opinion. A critical and informed expert opinion is preferable to a monkey throwing a dart, but it is still an opinion. It's also not easy, which is why systems analysts get away with charging exorbitant consulting fees.
The big issue that I've spent so much time introducing is the concept of a partially ordered set. In a partially ordered set, two non-equal elements can't always be neatly compared. Let's go back to the Buick example. Buick A costs $19,000 and Buick B costs $19,500. A is cheaper, so we should buy Buick A if the only thing we care about is the cost of the car. This is a fully ordered set, but how realistic is it?
A businessman once told me that the smartest thing you can do for a customer who cares only about cost is to refer him immediately and enthusiastically to your closest competitor. The implication is that cost is only one factor to consider when purchasing an automated material handling system. Just like a material handling system, the cost of your family car is important. But a cheap car isn't all that great if the wheels fall off as soon as you drive off the lot. So let's consider the mileage of the vehicles.
Suppose that Buick A has 50,000 miles on the odometer to go with the $19,000 price tag. If Buick B costs $19,500 and sports 5,000 miles, it's quite likely that you would happily pay an extra $500 to get a vehicle with fewer miles. But what if Buick B costs $19,500 with 45,000 miles on the odometer? Now you have a decision to make, because the Buick problem of cost and mileage involves two distinct elements of a partially ordered set.
The simplicity of Version 1.0 is a great strength. A novice instructor could easily assign any sample of student work into one of the three categories in this first incarnation of the Sinkhorn Rubric. The Version 1.0 rubric does not imply that two students who earn a C have turned in identical work. What the rubric implies is that each student who earns a C did well enough to pass but not well enough to earn an A.
For next time
I doubt any dean or chief academic officer would be fond of a grading scale with only A, C, and F, so I will expand the Sinkhorn Rubric to allow for a B and a D. If you have something to add to the discussion, feel free to use the comments below. I will decide on the numbering scheme for my versions and get back to you next time.
In the meantime, let me ask you a question. What is the difference between a B and a B+? I don't want to know the numerical difference. What I'm getting at is the qualitative difference between B and B+. What does a student have to do to get a B+ that is significantly better than a B but not worthy of an A or an A-? Would an employer who hired a B+ student of accounting expect significantly more than that of a B student?
One of my all-time favorite quips goes a little something like "If you think your teachers are tough, wait until you get a boss." Too true. You see, all good teachers need to be patient with learners. As I mentioned in my last post, Yoda was patient with Luke Skywalker. Purposefully, I did not go into great detail as to why great teachers are patient with their students. My purpose was that I did not wish to steal any thunder from this blog entry.
The link here between Yoda and evaluation of students is that honest, constructive assessment of student work requires many things: among them are clear expectations, acceptance of creativity, and patience. And the most important of these is patience. Students, like children and employees, benefit from high expectations. But it is counter-productive to insist that an instructor's evaluation of a student does not include at least a certain level of opinion.
In this first post, I discuss the biggest concern I have with evaluating student performance. I will beg your indulgence, since I take a longer time than usual to get to the point. So please, be patient.
But first, we need to talk about your flair
If you have never seen Mike Judge's feature length directorial debut, you should stop reading this and watch Office Space as soon as safety and decency allow. In the movie, Jennifer Aniston's character, Joanna, is a waitress working in one of those restaurant chains. You've been there. It's the place where members of the waitstaff are evaluated solely on their willingness to behave in the most obnoxious and ingratiatingly sycophantic manner allowed by law.
The emblem of this behavior is flair. In the Mike Judge world of a Chotchkies restaurant (Judge himself plays Joanna's supervisor), servers are required to wear at least 15 employee-supplied buttons-- the so-called flair-- on their uniform suspenders. "People can get a burger anywhere. They come to Chotchkies for the atmosphere." Suspender flair is presumably a big part of this atmosphere.
Joanna thinks, correctly, that the "atmosphere" of Chotchkies both debases their employees and insults the intelligence of their customers. Stan, the supervisor character, picks the wrong day to criticize Joanna for wearing only the minimum 15 pieces of flair, and her response is honest, priceless, and a textbook example of a person railing against a divisive and poorly designed system of assessment.
Joanna - "You know what, Stan? If you want me to wear 37 pieces of flair, like your pretty boy, Brian, over there, why don't you just make the minimum 37 pieces of flair?"
Stan - "Well, I thought I remembered you saying you wanted to express yourself."
Joanna - "Yeah. You know what? Yeah, I do. I do want to express myself, okay. And I don't need 37 pieces of flair to do it."There may be children who read my blog, so I won't include a video of what happens next, but picture Jennifer Aniston dressing up as Stone Cold Steve Austin for Halloween.
Suffice it to say, Stan is not a born leader. Mike Judge's character is a person who apparently got ahead by scoring high on the poorly designed Chotchkies rubric of evaluating employees. The Chotchkies rubric does not identify and reward leadership, since leadership is apparently not something valued in the mid-level management at this particular establishment.
Guerrilla Grading
In our accreditation-driven education system, exhaustively detailed grading rubrics have become a holy grail of sorts. In my opinion, it is extremely important that grading be both as fair and objective as possible. Thus, a rubric is a natural means of establishing a standard to which students can compare their work as a part of their own self-assessment.
This is a good thing for students and instructors. But I like to remember that professional educators are, well, professionals. And one of the most important things that all professionals share is the need to operate with some degree of autonomy. Simply put, educators earn a living at least in part because they know a good paper when they see one.
For me, the difficulty in grading student work is not determining right answers from wrong answers-- or even flawless logic from an argument that needs to be refined. My difficulty is in assigning points. More properly, my difficulty is in deciding how many points to subtract for a specific error or omission. Is a sign error a one point mistake or a three point mistake? The answer to that depends on the problem. Suppose I ask for the roots of x^2+4, a sum of perfect squares.
This is what I'm looking for.
If I instead get the following, the difference is merely a sign error. But it is exactly what I don't want.
A question of this nature is specifically designed to test if a student can recognize that a certain quadratic equation has no real roots. And I want the student to do this without having been provided the graph. Students can discover this algebraically by solving the equation, visually by plotting the graph themselves, or intuitively by recognizing that x^2+4 is a vertical translation of x^2.
The Sinkhorn Rubric[TM] - Version 1.0
As it turns out, life is a pass/fail course. According to Jorge Cham, this is also a reasonable characterization of life in graduate school.
The Sinkhorn Rubric is based on the pass/fail system. A Jorge Cham pass/fail system is kind of like getting either a C or an F. For me, an A is good, a C is good enough, and an F is not good enough. The first and most basic version of the Sinkhorn Rubric is as follows.
A - The student successfully completed the assignment with a substantive addition of desirable elements including, but not limited to, at least one of the following: clarity, brevity, insight, professionalism, innovation, or creativity.
C - The student successfully completed the assignment with no catastrophic errors or significant omissions.
F - The student did not complete the assignment, failed to complete a significant portion of the assignment, or did not provide significant justification for his or her conclusions.At Chotchkies, Version 1.0 would look like this.
A - A full 37 pieces of flair. And a terrific smile.
C - The minimum 15 pieces of flair.
F - Flipped off the boss. And a line cook who just happened to be standing there.In my studies of student success in courses like college algebra and developmental math, the DFW rate is considered one of the most important performance measures. The DFW rate is the fraction of students attempting a course who do not complete the course with at least a C. That is, they earn a D, an F, or withdraw. Consequently, I submit for your consideration that there is a peer-reviewed precedent of sorts for Version 1.0.
Beer, exams, and beauty contests
In a previous life, I was a blue-ribbon homebrewer. My summer wheat took first in the specialty category at the Kentucky State Fair many moons ago. I also was a beer judge as well. Judging homebrew is kind of like reviewing wine for a magazine, but homebrewers will drink pretty much anything that is both non-toxic and made with malted barley.
 |
| http://www.homebrewersassociation.org |
In the Homebrew division of the Kentucky State Fair, there are three rounds. In round one, each beer is judged against a style recognized by the American Homebrewers Association (AHA). Points are awarded based on how well the entry adheres to the style guidelines in categories such as flavor, bouquet, and appearance. The X number of beers with the X highest scores move on to round two.
Round two is the medal (ribbon, actually) round. Judges in each category decide on their three favorite beers, then arrange the top three in order of first, second, and third. In round three, the blue ribbon winners in each category compete for the coveted Best in Show award.
You see, round one is like grading exams. You have to decide what to take points off for, and you also decide how many points to take off for each mistake. Even though the AHA puts out every effort to make the process as objective as possible, it is still highly subjective. In fact, all judges in each category of round one are required to make certain that their scores fall within a certain tolerance of each other. And just like grading exams, judging beer for long periods of time has a tendency to dull the senses. For some strange reason...
Rounds two and three are like judging a beauty contest. In round two, all judges in, for example, the light ale category get together and decide which of the beers in their category are the three best. These are the ribbon winners. Then the judges place the ribbon winners in order of preference-- first, second, and third place.
Now let me ask you, my gentle readers. Would you rather judge round one or round two? That is, would you rather grade papers or judge a beauty contest?
AC/DC and fully ordered sets
From a mathematical perspective, the major difficulty associated with schemes for scoring homebrew entries is that the scores are generally whole numbers. As a subset of the real numbers, whole numbers are what is called a fully ordered set. That means that whenever two numbers in a fully ordered set are compared, either the numbers are equal or one is larger than the other. So, whenever two distinct (i.e. non-equal) whole numbers are considered, one of them must be larger than the other.
The simplicity of a fully ordered set is quite beautiful to a decision maker. For example, consider that you wish to buy a 2010 Buick LaCrosse. If Buick A costs $19,000 and Buick B costs $19,500, then Buick A is less expensive than Buick B. You will never need an expert to tell you that a $19,500 car is more costly than a $19,000 car.
And that's the beauty of a fully ordered set. But as soon as there is a matter of opinion, things get a little more dicey. We can easily look up the numbers to see that Australian rock band AC/DC's 1981 album For Those About to Rock reached a higher maximum position on the Billboard album charts than Back in Black. For Those About to Rock reached number 1. Back in Black did not reach number 1 despite becoming the second best-selling album in history.
But if you sit down and talk to any serious (or even a casual) AC/DC fan, you will be in for a long discussion of which of these is the better album. An album grading rubric might help, especially if all manner of AC/DC experts were involved in the creation of this rubric. But the determination of which is the better album ultimately must be, by nature, a matter of opinion. That's why music departments teach Music Appreciation instead of Music Scoring and Ranking.
Partially ordered sets and a cheap, used Buick
When the rubber hits the road, any kind of sophisticated decision comes down to a matter of opinion. A critical and informed expert opinion is preferable to a monkey throwing a dart, but it is still an opinion. It's also not easy, which is why systems analysts get away with charging exorbitant consulting fees.
The big issue that I've spent so much time introducing is the concept of a partially ordered set. In a partially ordered set, two non-equal elements can't always be neatly compared. Let's go back to the Buick example. Buick A costs $19,000 and Buick B costs $19,500. A is cheaper, so we should buy Buick A if the only thing we care about is the cost of the car. This is a fully ordered set, but how realistic is it?
 |
| http://www.autoinsane.com/2009/07/28/reviews/first-drive/first-drive-2010-buick-lacrosse/ |
A businessman once told me that the smartest thing you can do for a customer who cares only about cost is to refer him immediately and enthusiastically to your closest competitor. The implication is that cost is only one factor to consider when purchasing an automated material handling system. Just like a material handling system, the cost of your family car is important. But a cheap car isn't all that great if the wheels fall off as soon as you drive off the lot. So let's consider the mileage of the vehicles.
Suppose that Buick A has 50,000 miles on the odometer to go with the $19,000 price tag. If Buick B costs $19,500 and sports 5,000 miles, it's quite likely that you would happily pay an extra $500 to get a vehicle with fewer miles. But what if Buick B costs $19,500 with 45,000 miles on the odometer? Now you have a decision to make, because the Buick problem of cost and mileage involves two distinct elements of a partially ordered set.
The simplicity of Version 1.0 is a great strength. A novice instructor could easily assign any sample of student work into one of the three categories in this first incarnation of the Sinkhorn Rubric. The Version 1.0 rubric does not imply that two students who earn a C have turned in identical work. What the rubric implies is that each student who earns a C did well enough to pass but not well enough to earn an A.
For next time
I doubt any dean or chief academic officer would be fond of a grading scale with only A, C, and F, so I will expand the Sinkhorn Rubric to allow for a B and a D. If you have something to add to the discussion, feel free to use the comments below. I will decide on the numbering scheme for my versions and get back to you next time.
In the meantime, let me ask you a question. What is the difference between a B and a B+? I don't want to know the numerical difference. What I'm getting at is the qualitative difference between B and B+. What does a student have to do to get a B+ that is significantly better than a B but not worthy of an A or an A-? Would an employer who hired a B+ student of accounting expect significantly more than that of a B student?
Tuesday, July 26, 2011
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.
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.
Friday, July 22, 2011
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.
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.
Wednesday, July 13, 2011
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.
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.
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.
Friday, July 8, 2011
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.
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.
Thursday, July 7, 2011
I love it when a plan comes together
Welcome
Welcome to The Joy of Teaching... Algebra. Now that I'm back from visiting family over summer break, it's time to get serious about getting ready for next semester. In addition to a research project or two and prepping my tenure application, I'm working on redesigning my College Algebra course.
Picking a textbook
The first step of the redesign was picking a new textbook. That book is College Algebra: Real Mathematics, Real People, 6th Edition by Ron Larson. My previous text was College Algebra by Barnett, Ziegler, and Byleen. The Barnett text is good to prepare students for trigonometry and calculus, which completely missed the typical student in my class.
At Peru State College, students who do not test out of the math requirement in general education usually end up taking either College Algebra or Intermediate Algebra. Which means my College Algebra course is the only non-statistics course that many of our students take in college. And the Barnett text almost completely misses what I want my students to take out of college-level mathematics.
The ten-year test
One of the first questions I ask myself when designing a course is the Ten-Year Test[TM] question. Specifically, what do I hope students remember about my course in ten years? Now that I've finished my last degree about 8 years past, here are the results of some ten-year tests I've given myself.
Writing I - Pluralization (they vs. he and/or she)
Speech - Informative speech outline (Introduction, 3 main points, conclusion), indifferent feedback
Engineering Economics (Finance for you business types) - The time value of money, depreciation
Topology - Urysohn metrization theorem, open and closed sets
Human Factors - Radial and ulnar deviation, ischial tuberosity
In College Algebra, I want students to remember functions and mathematical modeling. But the Barnett text doesn't explicitly introduce functions until Chapter 3-- halfway into the semester. It is as though the ins and outs of operations on complex numbers are more important than actually establishing the intuition of a well-defined function. You won't win any contests guessing where I stand on that debate.
Planning the semester
As I've progressed in my career (this is year 8 of full-time college work, by the way), I've decided that I like having a somewhat detailed plan at the outset and deviating from it as needed. Kind of like the Miles Davis Quartet instead of Ornette Coleman-- except that those guys have actual talent.
At any rate, my first order of business was to draft the following weekly plan for the semester. Features include a non-standard amount of time between exams, two variable linear systems immediately after linear functions (mostly so I don't run out of time at the end), and a decision to hit only the bare bones of algebraic analysis of polynomials. The biggest weakness I see is the possibility of running up against the end of the semester right in the middle of logarithms which seem to require more sink-time than they might get. On the other hand, testing so soon after introducing the concept might help with that.
Welcome to The Joy of Teaching... Algebra. Now that I'm back from visiting family over summer break, it's time to get serious about getting ready for next semester. In addition to a research project or two and prepping my tenure application, I'm working on redesigning my College Algebra course.
Picking a textbook
The first step of the redesign was picking a new textbook. That book is College Algebra: Real Mathematics, Real People, 6th Edition by Ron Larson. My previous text was College Algebra by Barnett, Ziegler, and Byleen. The Barnett text is good to prepare students for trigonometry and calculus, which completely missed the typical student in my class.
At Peru State College, students who do not test out of the math requirement in general education usually end up taking either College Algebra or Intermediate Algebra. Which means my College Algebra course is the only non-statistics course that many of our students take in college. And the Barnett text almost completely misses what I want my students to take out of college-level mathematics.
The ten-year test
One of the first questions I ask myself when designing a course is the Ten-Year Test[TM] question. Specifically, what do I hope students remember about my course in ten years? Now that I've finished my last degree about 8 years past, here are the results of some ten-year tests I've given myself.
Writing I - Pluralization (they vs. he and/or she)
Speech - Informative speech outline (Introduction, 3 main points, conclusion), indifferent feedback
Engineering Economics (Finance for you business types) - The time value of money, depreciation
Topology - Urysohn metrization theorem, open and closed sets
Human Factors - Radial and ulnar deviation, ischial tuberosity
In College Algebra, I want students to remember functions and mathematical modeling. But the Barnett text doesn't explicitly introduce functions until Chapter 3-- halfway into the semester. It is as though the ins and outs of operations on complex numbers are more important than actually establishing the intuition of a well-defined function. You won't win any contests guessing where I stand on that debate.
Planning the semester
As I've progressed in my career (this is year 8 of full-time college work, by the way), I've decided that I like having a somewhat detailed plan at the outset and deviating from it as needed. Kind of like the Miles Davis Quartet instead of Ornette Coleman-- except that those guys have actual talent.
At any rate, my first order of business was to draft the following weekly plan for the semester. Features include a non-standard amount of time between exams, two variable linear systems immediately after linear functions (mostly so I don't run out of time at the end), and a decision to hit only the bare bones of algebraic analysis of polynomials. The biggest weakness I see is the possibility of running up against the end of the semester right in the middle of logarithms which seem to require more sink-time than they might get. On the other hand, testing so soon after introducing the concept might help with that.
| Week | Sections | Notes |
| 1 | Chapter P | Group work at board, work through review exercises, p. 68 |
| 2 | 1.1-1.3 | Graphs, lines, functions |
| 3 | 1.4, 1.5 | Graphs of functions, transformations |
| 4 | 1.6, 1.7 | Operations on functions and inverses (may take more time) |
| 5 | Exam 1 | |
| 6 | 2.1-2.4 | Linear equations, graphical methods and complex numbers (All but 2.3 are simple, 2.2 and 2.3 will be done quickly). Quadratics (2.4) will probably not be done by next week |
| 7 | 2.5-2.7 | Maybe skip or abbreviate 2.5 Solve other functions algebraically, 2.6 is inequalities, 2.7 is linear models and scatterplots |
| 8 | 5.1, 5.2 | 2D linear equations |
| 9 | Exam 2 | |
| 10 | 3.1-3.3 | Polynomials and the Fundamental Theorem of Algebra |
| 11 | 3.4-3.6 | Rational functions, asymptotes and graphs |
| 12 | 4.1 | Exponential functions |
| 13 | Exam 3 | |
| 14 | 4.2, 4.3 | Logarithmic functions and properties of logs |
| 15 | 4.4, 4.5 | Solving exp and log equations, exp and log models |
| Final | Exam 4 |
Conclusion - Just like Speech class
So that's the starting point for the All-New College Algebra at Peru State College. As I add to my blog, I'll report on my own efforts and the results of working with Profs. Reed and Young whom I met at the Mathematics Inquiry Based Learning Workshop at Ann Arbor last May.
Thanks for reading, and remember the summer weather when you get snowed in in a few months.
Subscribe to:
Posts (Atom)