Drama and Teaching

Great writing inspires. I have been reading over the last several months a small book that I have found very inspiring. Everytime I pick it up I get new ideas, and I challenge my assumptions about teaching. The most surprising thing about it is the book is not about math, not about teaching, not about students. It is about drama. It is called The Three Uses of the Knife by David Mamet. It speaks to the nature and use of drama.

I know that there is a general sense among many teachers that class does not need to be a show everyday, and clearly that attitude taken to extremes creates teachers that are more like Michael Scott of "The Office" than Dan Meyer.

But Mamet forcefully explains that humans have a dramatic urge. People want to see themselves as part of a giant play with themselves as the hero. As teachers we have a responsibility to engage students, and to ignore this paradigm seems short-sighted at best.

Mamet is very insistent that there are good forms of drama and bad forms of drama. He uses school, politics, the evening news as examples of bad drama carried out every day.

In the next few posts I will reflect on certain sections on Mamet's work and describe what I think it means for teaching. We will start the section he calls The Perfect Game.

Imagine the perfect game. Is it your favorite team thumping their opponent? Of course not, it is your team starting out well, looking dominant and then all of a sudden things fall apart. They change many things and nothing seems to work. Then all of a sudden when things look darkest, a player redeems himself and for some previous screw-up in the game, and scores the go-ahead score, but then the referee calls it back, and again the players must find a way to take the lead, and they do, but the other responds to retake the lead, only to have a more miraculous play occur for your team, and so on and so on.

This is the prototypical three act structure. Humans look for this structure in their lives all the time, and if you watch sports, or follow politics you will see that these activities are recast by commentators in this framework all the time. Mamet would tell us that this recasting is for our pleasure.

Mamet describes this as "Yes, No, But Wait...", and it repeats again and again.

You can imagine this in a classroom, students come in an see a problem something familiar that they know and can deal with. They deal with it easily and look to you for their deserved praise. Then you give them another problem similar, but with some unexpected twist. They work forward trying many things. Many of those ideas fail, and the way seems lost. You don't help them (much, if any) then at some moment through an idea spread through the class based solely off of students previously failed idea (or through some cryptic magic incantation you said under your breath) and the students are off and they solve the problem, or they don't stymied by some other detail they have left out, or you have given them a new problem with a new twist. In this way, we model "Yes, No, But Wait...". And basically we should believe that this is what students want, crave, desire. They do not desire the answer, the algorithm, the process. They desire to be heroes.


Part of what Mamet talks about is our never ending need to dramatize our lives:

For we rationalize, objectify, and personalize the process of the game exactly as we do that of a play or drama. For, finally, it is a drama, with meaning for our lives. Why else would we watch it?

That idea about rationalizing, objectifying, and personalizing is so important. Students need to do that to really connect with the content that we are looking for them to master. If they don't do it this more complex and deeper level, then they will never remember it.

Determinants as proportions

This is a continuation of a discussion of determinants started here. And there is more here.

Proportions

So if determinants are important how important are they and when can they be introduced reasonably. One place that they could possibly come up is in reference to proportions. As a reminder, a proportion is equation where two ratios are equal. One method to solve equations such as this is the Means-Extremes Property. This is more commonly known as “Cross-Multiplying”. I think I speak for many teachers that cross-multiplying is a “bane of existence”. Cross-multiplying is a rule that is often overused. It seems to quickly rise to the top of all students list of favorite methods so that whenever in doubt about how to proceed in a problem with fractions teachers often here the idea put forth that the correct method might include cross-multiplying. This is probably because few students really understand what this method does or why it works

Connections between determinants and proportions

So what is the connection between determinants and cross multiplying. Well
it can be seen from a variety of ways. First is in the formulas themselves. A
proportion has the form:

after cross-multiplying we know that

this last equation can be rewritten in terms of determinants as

In a proportion we are given that two fractions are the same. Each of those fractions can be thought of as vectors. Similar to the definition of slope as a fraction or as a vector. With this definition of the fraction we see that the two fractions will be equivalent if their vectors point in the same direction. If they point in the same direction then the area given by the determinant will be zero.

Advantages of the Determinant Formulation

The advantage of this determinant method to solving proportions is that we eliminate the fractions from the problem. Cross-multiplying would only exist in determinants, where the rightly do play a role. Students would be less likely to misapply the idea of cross-multiplication to every situation with fractions. Determinants would be introduced earlier and their presentation of area would be well supported. Students would also have to have a clearer understanding of slope as a primary way of looking at fractions.

Classroom Projects

So late one night last year, I had a strong desire to change the way I teach. In many ways I see myself as very traditional. Some people tell me not so much, but I think at least philosophically I am very much in the land of I have knowledge; their minds are empty; must put my knowledge in their heads. Despite this I definitely see myself as a constructivist. A bad word to many I am sure (I know the spell check doesn't like it).

So about a year later, I am doing something with my late night ponderings. In my 8th grade Algebra 2 class, we are doing a unit on quadratics with an introduction to complex numbers thrown in for good measure. I have done a few teaching to the whole class days, but mostly we have days for the kids to work on a variety of projects. Some examples you ask? Why sure

* Hardy-Weinberg Equations from Biology
* Understanding how complex numbers increase the range of quadratic functions
* Deriving the quadratic formula
* How do the a, b, c in ax^2+bx+c=y affect the graph of the function
* Real-life applications of parabolas

There are more, but you probably get the drift. Each student will have to make a "presentation" of some kind. Not every kid can make an oral presentation to the class, we don't have the time. I am hoping that technology will come to my rescue, and some kids will make little videos that I can assign for homework. Students will have to critique each other's work. These are teaching problems I haven't worked out before, but I am enjoying it so far...

Determinants are calling me...

Good Story about a class, and my continuing fascination with determinants.

I go to class today, and it is Algebra 2, which I am never really sure how to approach. We are doing a chapter on lines. Now I had all this students last year for Algebra I and it is an honors course, so they are all pretty strong. I come in to class and put the following question on the board:

Determine if the points (-2, 1) (0, 3) and (7, 12) are collinear

And do you know what all the students asked (Go ahead test yourself, see if you can put yourselves in the mind of an 8th grader)

I hope you guessed "What is collinear?" Cause that would be right.

Now we are at the tony private school, so the smartboard is still on from the last class. I login in and go to wolfram.mathworld.com and look up collinear. I let the students read the first couple lines, and now they are off and running. Get lots of good answers/approaches. Some students find the equation of line using two points and check the third point in the equation to see if it works. Some calculate the slope two different ways and see if they are equal. Some calculate the equation two different ways and see if they are equal. It is all very good.

While the students are working I can help but see the following information just a little bit lower down on the mathworld entry for collinear (and this is not a direct quote)

You can check if three points are collinear if with determinants. For points (x_1, y_1), (x_2, y_2), and (x_3, y_3) create the determinant.

|  x_1  y_1  1 |
|  x_2  y_2  1 |
|  x_3  y_3  1 |

if the determinant equals zero then the three points are collinear.

I had totally forgotten about this, even though I had written a post about determinants two months ago.

Now I am standing in front of my class while they work the collinearity problem wondering, should I go for it. Should I show them determinants right now? Should I use this moment to introduce this strange monster of mathematics? It then occurs to me that we can use this to find equations of lines as well, and that in the next chapter I am going to introduce Cramer's Rule. Cramer's Rule is nothing but determinants, and so I go for it. The kids really liked it, and it is a most amazing thing that the series of calculations does this.

The way to find the equation of the line is to take two points and create the determinant:

|  x    y    1 |
|  x_1  y_1  1 |=0
|  x_2  y_2  1 |

this works because you are looking for all the points that are collinear with the other two.

Another way to look at it is the fact that the determinant

|  x_1  y_1  1 |
|  x_2  y_2  1 |*.5
|  x_3  y_3  1 |

Gives the area of a triangle with those coordinates therefore the three points are collinear if and only if the three points define a triangle with zero area.

So the class went well, and they really had an appreciation for determinants. They liked the way this got them pretty close to standard form. They may even remember it. I showed them two by two determinants as well, we will see if they recognize it when it comes up in about two weeks.

So, after another foray with determinants, I left pondering. I wonder "What is a determinant?" What is it's core? It does so many things: linearity, area, invertibility, etc. I just don't know.