When Subtraction Problems Aren’t Really About Subtraction

There’s a subtraction problem making it’s way around the internet, supposedly authored by a electronics engineer / frustrated parent (but who knows whom the real author is…) which rails against the “Common Core Mathematics approach” to subtraction:

Online debate with math folk I follow on Twitter has centered on this parent’s misunderstanding of Common Core; that while the standards describe skills students should master, they do not suggest methods for reaching the standards.  My friend Justin Aion has done a fantastic job of summarizing the disconnect on his blog.  Feel free to leave here for a few moments are check out this wonderful summary.  Meanwhile, Christoper Danielson has provided an annotated version, which lends some clarity to the intent of the given question.  His blog provides additional clarifying examples.  It’s another great read!

To this engineer / parent, I would ask why, if time efficiency is your primary motivation, then why bother even writing down the problem?  If it takes you 5 seconds with your traditional method, and I can perform the calculation in 2 seconds with a calculator, then you should be fired and replaced as I can do the problem with a 60% efficiency increase.  But I suspect this parent doesn’t want to entertain this argument.

THE HIGH SCHOOL PERSPECTIVE

I’m a high school math teacher, so why should I care about the method students use to subtract numbers?  Once students learn how to subtract, they take a test to prove their learning, and we move on to the next idea.  It’s just that easy…..or is it?  Actually, this debate matters quite a bit to us high school folks, as it speaks to the global issue here: What does it mean to teach and learn mathematics?  Which of these choices best describes what you want from your math students:

• I want students to master a series of skills, and apply them when needed.
• I want students to understand structures, and apply these understandings to increasingly complex structures.

I suspect that most non-educators, and perhaps many of my math teacher colleagues, feel that the first choice is just fine.  And this is the problem.  Math in our schools is often presented as a series of disconnected skills.  Once you master addition, you get to do subtraction, then you can do multiplication, then eventually fractions and decimals.  4th grade math is a step up from 5th grade math, then 6th grade. Eventually, after we run out of those skills, you get to take algebra, which is a distinct experience from all previous maths.

It turns out that the skills students learn in elementary school, and their embedded understandings, have deep consequences when it’s time to consider algebra.  Students who have been exposed to methods which promote generalization, reflection on algorithms, and communication will find a transition to formal algebra a seamless experience.  Here’s an example:

ADDING RATIONAL EXPRESSIONS

Depending on your school’s or state’s algebra structure, a unit on rational expressions, and operations on them, often comes around the end of Algebra 1 or the start of Algebra 2.  A problem from this unit might look like this:

An activity I use at the start of this unit often allows me to identify students who understand structures, compared to those who have memorized a disconnected process.  Index cards are handed out, and students are asked to respond to the following prompt:

What does it mean to find a Least Common Denominator (LCD)?

I have given this prompt for many years, with students in all academic levels.  Without fail, the responses will fit into 3 categories:

• Students who use an example to demonstrate their mastery of finding LCD’s (such as: if the denominators are 6 and 8, the LCD is 24) without justifying which their approach works.
• Students who attempt to describe a method for finding a LCD, often using approproate terms like factors and products, but falling short of a complete definition.
• Students who don’t recall much about LCD’s at all.

Students are first exposed to least-common denominators at the start of middle school, perhaps earlier, when it is time to add fractions.  I suspect that many teachers support a similar approach here: make lists of multiples of each denominator, and search for the first common multiple.

It’s an effective method.  And many of my students, even my honors kids, have been “good at math” by mimicking these methods. But how many students can remove these training wheels, and can describe a method for finding an LCD given ANY denominator. Lists are nice, but they aren’t always practical, and they certainly don’t provide an iron-clad definition.  A procedure which ties together understanding of prime factors and their role becomes useful not only in middle-school, but carries over to our study of rational expressions:

A least-common denominator is the product of the prime factors in each denominator, raised to the highest power with which the factor appears in ANY denominator.

I don’t suggest that teachers provide this definition on day 1, and have students struggle with its scary-looking language.  Rather, generalizations require development, discussion and reflection.  Certainly start with making lists, but eventually teachers and students need to analyze their work, and consider the underlying patterns.

Here’s your homework: provide your elementary-age student a subtraction problem.  Then ask them to defend why their method works.  It’s the understanding of structure which seprates “skills in isolation” math versus “big picture” math.  We need more big picture thinking.