The Vicious Cycle of Algorithmic Learning

It seems like the harder we work at teaching students to be good at math, the more problems they have, and the worse they become.  We seem to be trapped in a vicious downward spiral.  I believe that a lot of the cause is that we’re trying to solve the problem at the wrong level.

The vicious cycle of algorithmic learning goes something like this:

1. Kids can’t do math.
2. We break the math down into small, easily taught and easily practiced pieces.
3. We teach the kids the individual pieces algorithmically.  The kids learn each algorithm as a kind of “muscle memory”.
4. We drill and assess the kids on each algorithm until they master it.
5. We remind the kids of each algorithm before the high stakes test.
6. Test scores go up.  We pat ourselves on the back.
7. Kids have no context for the algorithms, so they can’t remember them after the test.
8. Kids forget what they learned.
9. Kids can’t do math.  (And now we’re back to the top of the list.)

There are a lot of things we do learn algorithmically.  It’s useful for anything we need to remember in short-term memory.  Examples are your computer password or your girlfriend’s/boyfriend’s telephone number.  But if you’re like most people, if you try to remember what your password was three changes ago, or your girlfriend’s/boyfriend’s phone number from three serious relationships ago, you can’t do it.  You haven’t practiced the algorithm, so your brain forgot it.

Now pick up a pen or pencil and try to write the alphabet.  Of course you can do it—it’s easy!  You learned the alphabet algorithmically when you were in kindergarten.  However, the reason you can remember the alphabet is because you use the letters to make words, and the words to express thoughts.  Your brain has a huge amount of context surrounding writing letters on paper.  If you start writing the wrong one or forming the right one badly, your brain has a wealth of context that enables you to catch yourself immediately, correct the problem, and continue from where you left off.  The reason you can’t remember your old password or your old girlfriend’s/boyfriend’s phone number is because you have no more context in which you need those things.

Kids learn and retain math in the early grades.  There are always opportunities to count things, add, subtract, multiply, and divide.  Any kid with siblings knows how to apply knowledge of fractions to make sure he/she gets an appropriately-sized slice of cake.

However, by fourth or fifth grade, two things change.  The applicability of the math they’re learning to the world around them is less direct, and less constant.  The computational tasks are more complex.  To address these problems, teachers and curriculum authors break the complex tasks down into step-by-step, algorithmic instructions.  What’s worse, in order to isolate the task at hand, they strip it of its context.  Students learn the algorithms, but without context they have no way of predicting what kind of answer they should get, and therefore no way of assessing their own answers.  Their only way of sensing whether or not they succeeded is by the teacher telling them whether or not they got the right answer.

Without the ability to check their results, they have no way of applying the algorithm in any context outside of their math books.  The math they learn becomes just another password, which can be right or wrong, but which has no meaning outside the “password” box.  As soon as they learn a new password, the old one, which is no longer applicable to anything, gets discarded and quickly forgotten.

There is no “silver bullet,” but there are some changes that would help mitigate the problem.  One potential solution is to give students problems that build on prior learning and give them a chance to see how the problems they have learned to solve fit into newer topics.  (This used to be the norm, “back in the day”.)   Another solution is to give students problems that are more “context-rich”—in which there is significantly more information than is needed, and students need to determine which pieces are relevant.

In short, if we want our students to be good at math, we need to teach them how to use it.  If math is useful, they will learn and remember it.  But as long as it remains a bunch of disjointed algorithms, it is as useless as that old password that you no longer use—and just as memorable.

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