Pattern learning and why it hurts students

I work with a lot of students after school, many of who are struggling.  I always say it’s the best place to learn about issues students are having.  Students are more open to discussing their frustrations and you can watch a particular student work for much longer periods of time. I hear this from students (and parents) all the time: “My son seemed to understand it even when I went over the problems with him but he gets it wrong on the test.  I can’t understand it”.  So what is going on?

One of the first things I learned years ago is that students learn to solve problems by (smartly) recognizing patterns.  It’s tough at his age to have a real desire to understand “why something works the way it does” (I didn’t have it at 15 years old). But the problem with pattern learning is that it falls apart as soon as a problem looks a little different.

Example of common mistakes with distributive property

Let me give my first example that I see very frequently that involves the distributive property (of multiplication).

On the left is how students are typically are shown how to distribute. All of the problems they practice look like this so they learn to distribute whenever they see a number next to a parenthesis – without a thought as to why they should do it. But when they number appears on the right side of the parenthesis, they don’t know what to do (even though it is still multiplication and the 2 should be distributed).

Here is another example.  Students see the number 2 next to a parenthesis and they know what it should look like to distribute.  But there is a disconnect here because they aren’t thinking that it is multiplication (and therefore should be performed AFTER the exponent as shown on the right).  It is simply “something that you do when you see a number next to a parenthesis”.

Another great example is cross multiplying (I outlaw this in my class – students know they are not allowed to use this).  It is magic to them and they do to it because it got them the right answer on the “cross multiplying” quiz.  But they don’t know why it works and when they should use it because they learned it as a pattern – without understanding the math behind it.  So here on the left is someone cross multiplying and on the right is the same student trying to use it when it’s not allowed.

Finally the basic algebra problem.  They are used to solving equation in standard form (the x term to the left, the number to the right).  They can do a 100 of these problems but as soon as the format is changed, they fall apart:

The problem of pattern learning is compounded by teachers who give quizzes on particular topics but don’t mix up different topics enough.  Here is an example of a month in middle school:

1. Week one is multiplying fractions.  Quiz at the end of the week
2. Week two is dividing fractions.  Quiz at the end of the week
3. Week three is adding fractions.  Quiz at the end of the week
4. Week four is subtracting fractions.  Quiz at the end of the week.

This is a bit oversimplified to make a point, but generally students are taught topics as islands.  They exist for that week until the quiz is given, and then it’s on to the next (unrelated) topic.

What can teachers (and parents) do about this?

Here are some of the tactics we use to reduce pattern learning:

1. Mix problems types up on the same worksheet.  In the fraction example above, I would be sure week two worksheets had multiplying fraction problem mixed in with the division problems.  On week four, all 4 types would be mixed with the subtracting fractions problems.
2. Force students to explain why it works that way.  This might include explaining it to another student or to their group (when we do group work).
3. Spiral back old problems. We often either put old problem on quizzes weeks later, or we often create multi-part problems that require a part 1 that uses older topics, and then part 2 that needs that answer to solve the rest of the problem which has new material on it.
4. Create lessons and practice problems that illustrate why a rule works.  When we teach the exponent rule (same base – you add the exponents), we let them discover the rule on their own AFTER a couple of days of expanding each expression to solve the problem.  This is in lieu of giving them the formula up front and then teaching them to simply apply the formula. In the example below, the student does what I’ve shown on the right many times before every using the rule (on the right).