Tools and Applications

Abstract: good math problems can be used both to develop basic skills and to apply them. It is helpful to think about whether a problem is a good "intro" problem or an application problem.  Also, some things that we teach as tools may be better thought of as application problems for more basic tools.

This week I did a lot of thinking about my teaching philosophy.  I had the week off, and spent one day observing some other teachers in Salt Lake.  I also just read an article on problem solving for my teaching class, so those experiences are affecting how I look at things.

The basis of my thinking is that there are two parts to doing math: 1) having the tools to do math, and 2) actually doing the math.  To me, doing math is about making abstractions and connections.  This is often in the context of solving problems that you do not have a specific algorithm for.  Having the tools to do math means having some basic ideas to connect, and knowing how to process abstractions.

My problem with a lot of the high school math curriculum is that it seems to emphasize developing the tools without ever really doing any math with them.  Since there are a lot of tools to master, this is an important part of learning math.  But if you don't ever apply them to any higher order thinking, you haven't really learned math.  One result of this is that students often don't understand the tools they are working with and end up misapplying them.

So we have these two things: the tools and the connections.  I realized last night that when I think about these things I often imagine a process of learning tools and then applying them, that is, making connections with them to solve problems.  But there is another possible relationship: making connections to learn new tools.

In my observations this week I saw a lesson on solving systems of equations using elimination.  I was reminded of a presentation I saw at the Utah math teachers conference last fall.  A teacher introduced systems of equations by presenting the following situation:

Imagine a restaurant that doesn't post their prices.  One person buys two burgers and three shakes and is charged $5.  Another person buys two burgers and one shake for $4.

You probably don't even have to ask the question.  The students will almost immediately figure out the price for an individual burger and shake using something similar to the process of elimination.  You can then ask a slightly more complicated version:

Suppose the change their prices and someone now buys 6 burgers and 3 shakes for $12, and another person buys 3 burgers and 1 shake for $5.

As the questions get a little more difficult, the students end up using the rules of elimination that you are about to teach them.  Why would you do it this way?  It is interesting that students have some intuition about physical situations that mirrors algebraic skills, but when they learn the algebra they rarely go back and bolster their intuition by thinking of the physical counterpart.

My hope in developing tools through problem solving like this is that some of the physical intuition they have will transfer so that when they are doing algebra the right steps will seem natural, just like it does when thinking about burgers and shakes.  That is, I would like them to develop intuition about abstractions that is independent from, but aligned with their physical intuition.  I say independent because I don't expect them to always be reminding themselves of the physical connections.

In this problem, the students need some pretty basic tools: addition, subtraction, multiplication.  The purpose of the problem is to get them thinking in a way that will help them learn a new tool: solving systems by elimination.

After processing this way of presenting systems of equations I tried to make a connection with one of the most basic tools of geometry: the pythagorean theorem (PT).  Unlike systems of equations, students generally don't have much physical intuition to help them solve PT questions.  They either know it or they don't.  There are a variety of ways to derive the theorem that are accessible to grade school students, but none of them that I have seen provide much intuition about how to use the theorem.  One of the proofs involves using similar triangles, so it could be used a good extension of a unit on similarity.  But even that one is complicated enough that it is unlikely that any intuition about similar triangles would actually transfer to understanding of the PT.

This point is key:  the proof of the pythagorean theorem might actually be better used as a way to extend/apply understanding of similar triangles than to develop intuition about the pythagorean theorem. Some mathematical tools may not have useful "intro" problems.

Of course, another way to introduce the PT is to have students measure a bunch of triangles.  The relationship is just complex enough that they probably won't be able to conjecture about the nature of the rule unless they have already heard it, but it can be a good way to apply measuring skills and attach some physical intuition.

Another thing that we may want to consider is that there a bunch of things in our textbooks that are taught as basic tools (meaning stated up front like axioms and then practiced), but might be better off as applications.  An example is the extension of the PT to non-right triangles, known as the law of cosines:

a^2 + b^2 -2abcos(C) = c^2 (C is the angle opposite side c).  My students call the rule "tabasco".

This is actually a cool and very useful relationship.  But it is not something I want my students to memorize at this stage.  I don't really care if they have this as a basic part of their "toolbox".  I would rather they just be comfortable with the basic trig functions.  But a proof of this rule can be an interesting application of more basic trig and algebra skills.

So, to sum up, I think we need to develop an understanding of the relationship between our tools and our applications.  We should always be seeking ways to get our students doing math (higher order thinking), rather than just acquiring tools to be used in later classes.  But they also need to develop mastery of the basics.  One way to accomplish this is to do math strategically.  Before they learn a skill, give them a problem that they have intuition about to try and get them to transfer that intuition to the new abstraction.  After they learn a skill, give them problems that will require them to think about how (or whether) the skill can be connected to new situations.  In between, you may have a bit of drill and kill to build "muscle memory" type response.  But if that is all you do you will end up with a lot of misapplied tools.

The Growth Mystery (Warning: Graphs)

The following is a picture that has been haunting me for the past few days:

Another blog shows a graph that extends the phenomenon another 80 years into the past (although take this with a grain of salt since it shows no variation before 1890):

So what is it?  Basically, the graphs show US Real GDP (on a logarithmic scale) over time from the 19th century.  Surprisingly, despite all of the technological progress, drama, wars, and changes to our government during that time period, the growth rate hasn't changed.  The Great Depression? No problem, after some wild fluctuations, the economy returns to its regularly scheduled trend, as if it had never happened. It's as if history doesn't matter.

The first graph comes from the President's most recent budget, where he argues that the economy will return to normal trend just like it always has.  Of course, by this reasoning, it will return to trend regardless of what the President, or congress, or the Fed try to do.  It all doesn't matter in the long run.  GDP growth is driven by some invisible constant.

I am skeptical for two reasons.  First, it seems too magical.  Why would GDP growth remain constant?  Second, Real GDP is actually fake.  That is, nominal GDP (the amount in dollars) represents an actual amount of money.  But real GDP (RGPD)  represents nominal GDP (NGDP) corrected for inflation.  But I have little faith in measures of inflation.  How can you compare the value of a house with air conditioning to one without air conditioning, or this:

  with this: .

In an age of stagnant technology, real GDP might actually mean something.  But it is difficult to compare apples and oranges, and I think our life today is an orange compared the apple of 1890.  In fact, things are so different that the measure of inflation is even forced to change from time to time.  The standard measure of inflation is the Consumer Price Index (CPI) as measured by the Bureau of Labor and Statistics (BLS).  It is based on the prices of a "basket of goods".  But this basket includes things like airfare, computer software, and medical services that didn't even exist back then.  So how is that a realistic comparison?

I don't think that inflation is a totally useless concept, I just don't think it makes much sense for long term comparisons.  But the mystery of growth still remains.  If inflation and hence RGDP are made up, why do they end up giving us such a perfectly predictable growth rate?  Perhaps the way we calculate RGPD ends up reflecting something entirely different than overall productivity.  It might not make sense to say that the amount of value produced by the economy grows at 1.8%, but there must be something.

So, check out this next graph:

Population!  US population has grown at a constant rate of 1.3% annually since Europeans first arrived (I assume this graph doesn't include native populations living in the current US boundaries).  RGDP, as currently measured, seems to be perfectly correlated with population growth.  But wait, aren't the GDP graphs PER CAPITA graphs?  That is, GDP per person is growing at 1.8% and population is growing at 1.3%, so total GDP is actually growing at over 3% annually.  This is much more than the rate of population growth.  That means we can't simply say that productivity remains constant.  Still, it is something to think about.  Perhaps as the number of people connected by a single social system grows, the productivity of each person increases as well.  In any case, we now have two perfectly exponential graphs to deal with.  And how about a third?

Total energy use in the US is also growing at an exponential rate, about 2.9%.  This is quite close to the growth in total RGDP.  Also, notice that there is a drop off at the end in both energy and GDP graphs.  And it is not due to the recent recession.  The drop off in energy use started in 1980.  Plus, it is pretty obvious that energy use can't keep growing at 3% per year forever without some serious world changing technological advance.  If we took energy growth at only 2.3% (to make the numbers work out better), the US alone would be consuming all of the sun's energy within 400 years, and all of the energy in the galaxy within 2500 years:

That's a lot of energy.  But the fact remains that RGDP continued to grow at a healthy pace for a few decades after energy use started to drop off.  Maybe GDP used to be perfectly correlated with energy use until things started to get more energy efficient?  The efficiency of cars and trucks certainly went up over that time period:

Another thing to think about (are you sick of graphs yet?) is labor force participation.  It seems odd that per capita GDP growth would remain constant if population is growing at a constant rate, but the labor participation is not:

Notice the spike that happened starting in the 1970's.  Perhaps RGDP is correlated with energ use, but just before energy use started to decline the labor force participation rate started to go up.  This may have temporarily masked the effects of reduced energy use.  Now let me remind you.  If this story is true it implies that RGDP growth will almost certainly not follow the trajectory we have been on for the past several hundred years.  It will track energy use.  But this doesn't mean that we can't become more productive or lead better lives.  As I mentioned earlier, RGDP growth is a phony concept anyway.  We can't compare the value of items in an environment where technology is changing so fast.  Plus, if I am right about RGDP growth tracking something like energy usage + labor force participation, then advances in energy efficiency would show up as negative productivity growth.

In any case, it is likely that both energy use and population growth will follow something called a logistic curve.  These may appear to grow exponentially for a while, but eventually they start to flatten out:

So what do I make of all this?  Well, I think the conclusion the President draws from his graph is unrealistic.  Just because RGDP growth has been constant for the last 400 years doesn't mean it will keep growing that way forever.  It sounds funny to think about it that way, but that is how a logistic curve works.  Second, it may not be such a terrible thing for RGDP growth to level out because we can still enjoy any improvements that come from improved efficiency and technology.  RGDP doesn't seem to do a good job of measuring these things unless they are correlated with increased energy use.  Finally, my whole analysis could be way off.  I just started thinking about this seriously last night.  So I welcome your thoughts. 

Lion Cubs and the Game of Math

Let me compare learning math to a lion cub learning how to fight:

I am not sure they are having fun, but it looks like it.  In any case, they are not being forced to do it by their parents. It's a game that prepares them for life.  I think math should be a game.  But a lot of students aren't having fun.  Why not?  Well, the stakes are high, they don't "get it", and they don't have a choice.  Plus, different students prefer different kinds of games.

I just read a few articles for my curriculum class about differentiation.  One of them talked about different learning styles, and the other was about giving students "light bulb" questions, challenges they can do for extra credit. In terms of making math a game, both of these concepts seem promising.

But before I talk about the connection to games, I should say that I am a bit of a differentiation skeptic.  Do you believe it?  How can any teacher deny that students have different learning styles and need different approaches?  Well, I don't deny it.  I just have questions, doubts.

My reasons for doubting are: 1) if a student is weak in some area and strong in another it is not immediately obvious to me that we should avoid exercising them where they are weak, and 2) it is hard to plan different activities for different students, and I am having enough difficulty trying to decide on a single meaningful approach to each concept.

Ok, so let's put those concerns aside for a moment and talk about games.  What are the important features of an activity that make it a game?  I think a game is a bit of a paradox.  Games have to be challenging, but simple enough that you know how to play.  They have to provide clear feedback, but the stakes can't be too high.

One way that I have attempted to make math a game is by using Khan Academy.  I assigned a few homework assignments from the website, but in addition to homework I told the students they could get extra credit by getting more "points".  That is, the website keeps track of how much time you spend watching videos and how many questions you do and gives you points for it.  This was entirely optional for the students, the points give clear feedback about how you are progressing, and they can engage pretty easily.  There are some very basic modules at the beginning, like addition and subtraction, so you can get a decent number of points by "gaming" the system.  But I think that is okay, as long as they start to think of it as a game.

So far about two students from each class have taken up the challenge.  A few of them are engaged in a very serious points battle representing hours of doing all sorts of math problems online.  It is differentiated to a certain extent because students can do whatever modules they like (or just watch videos) and it is optional.  For those students participating, it is a game.  They are amassing huge numbers of points and badges, and they are proud of their accomplishments.  The only problem is that only a small percentage of students are participating at this point.  But it is a start.

Another project I am working on is even more explicitly about the game.  I made a "math dungeon" add-on level (called Euclid's Labyrinth) for a popular role-playing computer game.  The game as it exists already includes areas where the player has to solve certain puzzles to advance.  I am just changing the balance a bit in so that the puzzles are more challenging and more closely related to the curriculum I want to teach.  This is a work in progress, and it may not appeal to every student.  But teaching math in the context of a game is something I think holds a lot of promise.

However, this leads back to one of the problems mentioned earlier.  It may be possible to teach math to computer nerds in a role-playing game format.  And some RPG gamers actually do need a lot of math help.  But they are certainly not the only ones that need to learn math.  While I think high quality computer games could revolutionize math for some, the real breakthrough would be to make the classroom a game. (Although I do think computers will eventually replace many aspects of the classroom).

Giving challenging problems and making them extra credit seems like a promising idea because it allows us to give feedback in a way that seems to lower the stakes enough to make it a game.  I have tried this recently in my Algebra II class and the engagement rate is about the same as for the Khan Academy: one or two students per class.  This is probably a sign that I need to tweak my process.  But students seem overwhelmed, and they don't always like to take on optional work.

So how do I gamify my classroom?  I think part of the answer lies in choosing interesting problems and presenting them in a way that all students can engage (or maybe it is too much to expect every student to engage in a given game...).  Another part lies in finding a balance of giving meaningful feedback (like the Khan Academy points) in a way that they care about but doesn't stress them out.

Kicking Against the Pricks

Today was rough.  I introduced my students to chords (the kind in a circle, although the word originates from the Greek for harp strings) by having them draw circles, make some chords and verify that the perpendicular bisectors all go through the center.

The students in my last class broke two of my rulers and scattered the pieces.  I know I shouldn't let this kind of thing get to me.  Who cares about a few rulers?  But it just pissed me off that I have to put up with this kind of thing.  It made me want to get back at them somehow, which is pretty low.  Why should I want revenge on a 15 year old for breaking a ruler?  I don't even want to analyze why.  I am sure there is a good reason.

I came home and ordered a pizza.  Mercedes is in Arizona for the weekend so I just started surfing the internet.  No school next week so it feels like an uber-Friday.  I have a lot of work to do to get caught up but I can afford to take the night off.  I ended up reading about Rolling Stone's list of the top 500 albums of all time: The Beatles, Bob Dylan, Kurt Kobain...Somehow their lives all seem kind of tragic.  Even the ones that lived long lives and prospered.

Then I read about how why Ben Bernanke changed his mind on monetary policy at the zero bound, and why Judge Napolitano was fired from Fox News.

In case you were wondering, Bernanke was a brilliant monetary scholar that advocated some non-standard policies until he was put on the board of the fed, which has a strong culture of group-think.  Napolitano was fired shortly after he gave a rant about how both political parties exist mainly to promote big government. Regardless of wether we should adopt 4% inflation, or whether there really is a difference between Republicans and Democrats, it is hard to kick against the pricks.

My own recent experience with the administration has calmed down a bit.  They issued a letter telling me to adopt a more traditional teaching style, and then my principal observed my class a number of times to get a more concrete idea of how things are going.  Since then the conversation has been more conciliatory.  Still, it is tiring.

Ultimately, this job isn't worth it unless I have a sense purpose.  For me, "purpose" often translates into something controversial, which makes things hard.  This break will be a good chance to refocus.  How can I make this work?

Standard Based Grading

One of the things I have discovered recently is that students are sometimes more willing to engage if you call something a quiz rather than just having a discussion.  Students are overburdened.  They are looking for ways to reduced their workload and one way they do this is to focus their energy on performance situations.  I can take advantage of this by increasing the frequency of performance situations, hopefully in a way that doesn't increase their stress levels too much.  The perfect situation would be one where they think that their performance "counts" but where they are confident that if they try they will succeed.

So, after coming to this conclusion I read a number of articles and blogs about how we should abandon a traditional grading approach in favor of something called standards based grading.  That is, the grade book should just be a list of standards, and grades should be nothing more than a mark indicating whether a student has mastered that material or not.  One of the bloggers (I can't even remember who) said that she (I think it was a she) was done using grades as punishment or motivation.

There is a lot to like about this approach.  It makes a gradebook more transparent (although students may think a gradebook that includes a list of specific assignments is more clear). It helps students focus on what they need to learn.

One problem is that it can be hard to identify good standards.  It is easy to have a box in my gradebook for "sin, cos, tan".  Students can demonstrate that they know how to do opposite over hypotenuse and away we go.  But what do I put down if I want to test whether my students can apply problem solving skills to find the answer to a problem for which they don't have an explicit formula.  I am not really trying to test their knowledge, but their mathematical skill.  There may be more than one right way to solve the problem, or it might involve a number of steps and I want to test their persistence.

Why not just call it "Quiz #5" and use a grade as a motivator to get them thinking?

Teach Proofs?

Students don't like proofs.

In the last week I gave my students two different kinds of problems to solve.  One involved a complicated picture where the students had to use various ways to find the measure of angles that allowed them to calculate other angles, and so on until they found the measure of about 10 different angles:

The students engaged immediately, and continued working until they found the answer.  At first a number of them were confused about how to find the measure of the angles in the pentagon and hexagon, but they looked through their notes and found things to help them.  Not all of them found the correct answers, but there was a lot of thinking going on.

Yesterday I gave the students a different kind of problem:

The students shut down.  Why? I think the answer is feedback.  When asked to prove something, the students can't really identify when they have the "right" answer.  regarding a different problem, a student came up afterwards and said, "if it isn't true then the shape can't be a parallelogram, so it must be true."  Of course, this is how we proceed to construct a proof by contradiction, so I told him his intuition was good.  But he didn't quite understand that there was anything else to prove after making that statement.  The conclusion was more obvious to him than the theorems I wanted him to use: alternate interior angles, side-angle-side, etc.

In the first problem, they have a situation where they don't know a piece of information and they try to find it. In the latter problem, they take a piece of information that seems obvious to them and are asked to "prove" it using things that are much less obvious.

One of the objectives of my proof website is to get students to develop a visual understanding of when a proof is complete:

CongruenceProof.com

Still, I don't think this entirely solves the problem.  Students don't like proofs because the don't "get" proofs.  They don't see the point of an axiomatic approach to mathematics.

Proving things is the essence of mathematics.  Prior to geometry students would never suspect this because they aren't really asked to articulate their reasoning.  They simply learn algorithms.  An algorithm is different than a proof because it always proceeds in the same way.  A proof requires searching for connections.

Proofs are central to mathematics, but for most students a more important question is whether they are useful for anything else. The new common core suggests that they are not.  Deductive proofs are mostly replaced by a focus on transformations.  I happen to think that a focus on proofs develops very strong reasoning skills that transfer better to other applications than algorithmic skill.

I am currently undergoing a sort of writer's block.  I don't know what to say because I don't even know what my opinion is.  Should I give up on teachings proofs or should I continue to try to make them more accessible?

One idea is that I could focus on asking students to find fault with proofs instead of asking them to create their own.  Maybe this kind of troubleshooting approach would develop the kind of reasoning skills that would transfer to more general applications and prepare them for higher level mathematics without requiring them to buy in as much to the importance of alternate interior angles and other fuzzy seeming theorems.