Wednesday, October 16, 2013

Math Lit FAQ: Why are topics in the book ordered in that fashion?

The topic ordering of Math Lit, and all pathways materials for that matter, is decidedly nontraditional.  The traditional ordering of topics such as rational expressions in chapter 7 and writing expressions in chapter 2 is not the only correct ordering.  It's just a conventional one.  It's the one someone chose years ago in a textbook and others adopted.  The traditional order is linear, for the most part.  But there are some topics like proportions, variation, statistics, and geometry that seem to move around from textbook to textbook, always feeling not quite in line with the rest of the chapter.

The problem is that math is not completely linear.  Sure, there are topics which follow from each other in a particular order.  For those topics, we address them in a linear fashion in the book.  But many topics have great flexibility in where they can be explored.  And often, they make more sense when they are not in the traditional order.

So, back to the original question:  why are the topics in the book ordered in that fashion?

Answer:  Because it works.

To me, mathematics education isn't about "covering" topics.  It's about creating capable problem solvers.  To do that, we have to connect lots of areas of math.  Real problems are messy and not tightly defined as only an equations problem or only a geometry problem.  Real problems transcend multiple strands of math at once, requiring the student to integrate and apply.  So we develop the content in an integrated way, constantly applying what is developed.  The result may seem unusual at first, but only because we are so conditioned to a particular topic order.  That order isn't right or wrong; it's just familiar.

Consider this example:  In traditional beginning algebra texts, combining like terms is usually in chapter 1 or 2, in preparation for solving equations.  Later, in chapter 5, the exponents rules will be developed in preparation for multiplying and dividing polynomials.  These ideas are developed weeks apart and as separate ideas, but they need to addressed together to make sense of them.  So we develop like terms right around the time we're working on exponent rules to see if students really get when the rules apply and when they don't.  It leads to a much greater understanding of both topics.  And then we use both ideas in applied problems so that further connections are made.

We know these developmental students have not been successful in traditional algebra.  Doing the same content exactly as it was done the first time is not necessarily going to lead to different results.  Shaking things up and approaching content with problem solving in mind engages these students more and helps them make progress with their understanding.  It is exciting to move from a "check off this skill" culture to one where students are constantly making progress and capable of solving more involved problems.

It's also a lot more enjoyable to ebb and flow between topics, coming back to them regularly in a deeper way, after the mind has had time to work on other things.  This is quite different than taking a topic and going through every perspective of it all at once.  Students can get lost in the trees and forget there is a forest.

Does content being enjoyable matter?  Sure it does.  I don't believe my job is to entertain; it's to educate.  But students have to engage to learn.  And motivating them can lead to engagement.

At first, it may seem like a topic hasn't been "finished" because every facet has not explored before leaving it.  Give it a little time and it will come up again.  Students will be able to do more with it each time and increased understanding follows.  But it takes time and patience to see that.  Looking at three sections of any integrated text won't give the full picture of what it can help accomplish.

I think Einstein said it best:

"We can't solve problems by using the same kind of thinking we used when we created them."

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