Teach Math Like it's an Elective

Yesterday in my curriculum class we had a pretty good discussion on whether algebra is necessary for most Americans.  I still maintain that it is not.  In the Air Force, which I think can represent a good portion of middle class jobs, I needed to be fairly skilled in arithmetic.  But I never really needed to use variables or solve a system of equations using elimination.  Also, I think the majority of Americans can't do basic algebra, so it would be a surprise if it were absolutely necessary for their jobs.

My motivation for saying algebra isn't required for most jobs is to change the way we look at math education.  I think math is a great way to develop good logic and problem solving skills and I think it can be a lot of fun.  However, I don't think it is the only way to develop these skills, and aside from the fact that people use math tests (e.g. SAT) as a signal of education and intelligence, I don't think knowing any of the techniques beyond arithmetic is strictly necessary.

So my point is that in an ideal world we wouldn't treat math as if it were vocational education, like some manual that people have to master in order to do their job.  It isn't.  Anyone who isn't enjoying math is probably not getting much out of it.  Instead, I think we should treat math more like an elective.  That is, design our lessons as games, projects, experiments, etc. that are designed to get students thinking.  If this takes more time and we don't have time to drill a formula, so be it.  The formula probably isn't that important to someone who isn't going to major in math or science.  But as long as we think that math is a series of indispensable techniques and formulas, we can never justify slowing down because we might miss something.

Now, if someone wants to specialize in math or science, a lot of things become indispensable.   But I completely disagree that we are going to get more students willing to specialize in math and science by subjecting them to a death march through Algebra II.  Plus, the most important thing for people who want to specialize in this area is to develop certain habits of mind.  I personally thought math was a mindless chore until calculus. It was only tolerable because I was better than most of my peers.  I was lucky in that I had a great calculus teacher (and great teachers at the college level) who were able to open my eyes.

If we really want people to go into math and science, make math interesting.  Because it is interesting before we ruin it.  And we are ruining it.  I wish I could say I have been able to do this in my own classroom.  I haven't.  I still basically follow the prescribed curriculum at the recommended pace, and I get a lot of push-back when I try to do things differently.  Plus, I don't really have time to reinvent the wheel for every lesson.  So I feel like my teaching is a patchwork that doesn't really represent my still-developing philosophy.

Note:  one of my colleagues sent me a textbook from a school that seems to have done a pretty good job at implementing a curriculum more like what I imagine.   It is called the Park School of Baltimore.  I am now looking into it more closely as part of my class, and it seems like a lot of fun.  Check it out.

Grades

Read this rant about the pointlessness of grades.

The author is a recently retired physics teacher who says that grades have replaced learning as the primary objective in grade schools, and that they are not a good reflection of mastery. The author makes a lot of good points, but in the end I can't fully agree.  My main reasons for not hating grades are:

1) Students are actually motivated by them.  Grades are a way to transmit the necessity of learning.  In the end, we don't just learn because it is fun but because it is useful.  Students are not really in a position to know what skills will be useful so we have to tell them somehow.

2) Students are not very good at achieving mastery goals, so giving them credit for behavioral goals (which the author calls "completion credit") is a way to motivate them to take the small steps necessary to master necessary skills.  One interesting study in this regard involved making small payments to students.  When students were payed for long term, abstract objectives they were motivated but didn't really know what to do.  When students were payed for more concrete inputs (like the number of books they read)  they knew exactly what to do.

So, I agree that right now grades are pretty arbitrary.  Students and parents get stressed out about them and try to influence teachers.  They aren't that great at reflecting mastery.  But what would we have without them?  How could we indicate to students that learning is actually important?

Of course, this contradicts my view that a lot of things taught in school actually aren't important.  Maybe we should be more selective about the things we choose to grade.

Next item:  The Algebra Project.

The premise of this project is that everyone needs to have access to algebra because it is the gateway to more advanced math and science classes, and ultimately, college.  It is run like a grass roots organizations, trying to get struggling communities to pull together to support algebra in the way they came together during the civil rights era.

I don't think everyone needs to know algebra.  I don't think everyone needs to go to college.  Pulling a community together in support of math is a great idea, but it is also a utopian one.  The fact is that our economy doesn't really need people with basic algebra skills.  The economy needs people who actually really like math and science and can motivate themselves all the way through a graduate degree in engineering because on some level they think it is all very interesting.

Organizing a community to teach 8th graders algebra will have positive results, I am sure.  Hopefully a few of the students gain enough confidence to keep going with math and science.  And organizing a community seems like a better idea to me than passing a law like No Child Left Behind.  It seems more organic and less of a top down institutional solution.

But programs like this mostly just remind me of how messed up our system is at the very foundations.  We try to force kids against their will through a factory/day care system with an endless series of curricular goals with little evidence that these goals are actually useful.  Why?  Because this is what people need to get into college.  And why do colleges take high school so seriously?  Because if they just had placement tests they would have to admit a bunch of 12 year olds (honestly, a motivated 12 year old could easily surpass a typical high school graduate).  This would interfere with our idea of college as a coming-of-age party.

So why do we care about college?  Because employers take college seriously.  Again, why?  Employers are perfectly happy to let people waste 4 years in college because 25 year olds (especially ones that have been jumping through educational hoops for 20 years) are a lot easier to manage than 15 year olds.

Anyway, the point is that for most jobs you don't really need to know algebra, or anything else that is taught in high school or the vast majority of college courses.  High school and college just give us time to grow up.  I personally am not too upset at having spent the majority of my life in mostly useless educational settings because I actually enjoy learning.  But lets not be too idealistic about the impact of teaching kids algebra.

The Mormon People

A friend of mine recently completed a book entitled "The Mormon People."

I haven't finished it yet, but one idea in the book has really caught my attention.  Mormons are progressive.  That is, they believe in the power of institutions to improve people and society.  Once it is stated in this way, it seems obvious.  The LDS church is a behemoth that does a very good job at creating a standardized religious experience in nearly every country of the world.  And it seems to work.  Mormons are healthy and productive, and I don't think it is purely self-selection.

But I am skeptical of progressive ideals.  I tend to believe that giant organizations are out of touch.  Grand initiatives have unintended consequences.  Change is necessary, and efficient change is most often the result of "market" interactions, that is, free transactions that happen at the level of individuals.

However, I also realize that top down organization is efficient in some circumstances.  In fact, one of my favorite articles of all time is the Ronald Coase classic, The Nature of the Firm, in which he tries to analyze the conditions that determine how large an organization can get before the benefits of organization are outweighed by the costs.  The basic idea is that organization can reduce transaction costs at the micro level, but that decision makers become too far removed and inadequate to operate effectively.

In general, when there are two competing theories I think it is a good idea to ask ourselves: "are there conditions under which each theory is true?"  This applies to individualism vs progressivism, as well as classical vs Keynesian economics, and direct instruction vs discovery.

In my recent post on the purpose of the education system I called public education a progressive institution.  I meant this as a bad thing.  But surely there is an optimal mix of top down organization and local flexibility.  I suppose I just think we are erring on the side of utopian institutionalism.  Our ideal of No Child Left Behind is stifling, and I don't just mean the law that goes by that name.

Why Problem Solving Doesn't Work (According to One Author)

The following article makes a pretty strong case that direct instruction is the best way to teach:

http://www.aft.org/pdfs/americaneducator/spring2012/Clark.pdf

The basic idea is that when asked to solve problems that they don't know how to do, a student's working memory gets overloaded and they can't function well.  People solve problems best when the means for solving them is somewhere in their long term memory.  Experts can solve problems because they have a lot of schema in their long term memory, and each of these processes only counts as one item in their short term working memory when they pull them out to help solve a problem.

In other words, people solve problems most effectively when they already know how to solve them.  So the best way to get students to solve problems is by simply showing them how to do it.

This conclusion directly contradicts what I have been trying to do in my class, and I think it is important to occasionally question my premises.  But the author seems a bit too confident for my taste.  For example,  I think the relationship between long term memory and working memory is more complex than the author lets on.  In particular, the presence of a schema in long term memory is not either yes or no.  Students can "know" something in a very superficial way.  All of my students "know" the Pythagorean Theorem in the form a^2 + b^2 = c^2.  But many of them have a very hard time with the distance formula (which is equivalent to the Pythagorean Theorem), even if I suggest that they plot the points and make a right triangle.  This is probably because their working memory is overloaded by the process of plotting points, when what they really want to do is remember a formula.

What I see is a deficiency in being able to put together small steps to reach a goal.  I don't think that most students will ever even use the distance formula.  But it takes practice to have confidence making a small step, flushing your working memory, and seeing the problem anew.  That is, I am still a believer that the students need to learn problem solving itself, more so than they need to learn any one given formula.  In some professions it may be the case that all an expert must do is learn a limited set of algorithms and repeatedly apply them.  But what if we ever come across a situation that we have never seen before?

Having stated my objections to the theory of the paper, I should mention that there is actual research that shows that students learn better by direct instruction.  The theory is partially just an attempt to explain the data.  But I am equally skeptical about the data.  For one thing, education research is inherently hard to do.  The teaching methods used in the studies to represent discovery or problem based learning don't bear a very close resemblance to what I do in my class.  And the methods of measuring learning are not always flawless.  The design of the study also makes a huge difference.  In one case, researchers repeated one study that "proved" direct instruction was better with one key difference:  they continued to measure mastery over a longer period of time.  The results were diametrically opposed to the conclusions of the first study.  Over a longer period, the students taught by direct instruction forgot everything, and those who discovered the material slowly began to really understand.

I don't think this one study proves the case for discovery learning, but it should make us pause.  Research in education is hard to do, and it is always flawed.  This is unfortunate because it would be great to have some reliable data.   Theorizing by itself often leads to wrong conclusions.  At this point I am mostly convinced by what I have seen with my own eyes:  students who have been taught primarily by direct instruction are not very good at solving problems, and many of them are not very good at remembering what they have learned.  However, as mentioned in the article favoring direct instruction, problem solving in class can be very easily dominated by a few eager students and while the rest of the class remains passive and unengaged.

The End of the Education System

What is the end of the education system (as in the goal, not the apocalypse)?  From what I can tell, it is to ensure that every child in America is ready for college after graduating from high school.  For math teachers, that means that as a minimum they need to know the curriculum up to and including Algebra II.

The public education system is the most progressive institution in America.  By that I mean it is an effort by the government to achieve an obviously desirable social goal.  The problem is that progressive goals are often based on a simplistic view of reality, and they have unintended consequences.  I am coming to the view that the education system is based on some major misconceptions:  first, it is a good thing for every American child to achieve the same education; second, that it is possible for every American child to be given the same level of education.

One of the things that makes me question the whole system is the difficulty of teaching mathematics through problem solving.  The main obstacle is that a lot of students simply aren't interested in learning.  It is possible to give students a worksheet full of similar problems, tell them how to do it, and then make them work for an hour.  It is not possible to force students to engage in problem solving for any extended period of time if they don't want to do it.

School is mandatory.  Students are forced to attend, and they are forced to take math when they get there.  Thus, many of them are there against their will.  A few of them decide that even if they don't want to be there they might as well take advantage of the opportunities that exist.  Others protest.  You can make me sit here, they say, but you can't make me think.  And they are right.  We can't make them think.  And if they don't think, they won't really learn.  Sure, they will be exposed to a variety of mathematical techniques and they will acquire some familiarity with a variety algorithms.  But I think that the knowledge they gain is next to useless.  They won't be able to apply it because that would involve understanding, which requires thinking, which they refuse to do (at least some of them do).

The truth is that students may not know what is good for them, and they may be better off for having been forced to learn.  Unfortunately, the government doesn't really know what is good for them, either.  Is it really necessary for everyone to understand complex numbers?  I think not.  Most students would probably be better off learning some specific skill in some kind of apprenticeship.

High school is the result of flawed assumptions about what people need to know to be useful to society. The assumptions are flawed because they are the result of a political process instead of a "market" interaction between supply and demand.  No group of experts (and especially not state legislatures or school boards) is expert enough to understand a complex economy and know what skills are actually useful.  If we could know, the answer certainly wouldn't be that everyone ought to learn the same skills.

There is no demand for the kind of education we are giving aside from political demand.  Students don't want to learn what we are teaching.  The economy doesn't demand that they know what we are teaching.  Students often ask when they will ever use what we expect them to learn and for the most part the answer is never.  If they were actually interested in what we were learning they could make a career out of it (and a highly lucrative one).  But if they aren't interested they will never achieve the level of mastery required for the knowledge to be useful.

This perspective is a bit troubling for a public school teacher.  I basically don't believe in the whole premise of the institution that employs me.  I don't think every student needs to learn geometry or algebra to be successful, and I think it is nearly impossible to teach anything useful to students who don't want to be there.

The thing is, I like teaching math, and I like working with teenagers.  But I am pretty disillusioned about teaching math to teenagers who don't want to learn math.