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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:

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.

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At my high school, I host “math lab” on alternate days.  The lab is an open room where students can obtain math help during their lunch or free period.  I like this assignment because I get to see a lot of different kids during the week, gain some insight into the approaches of my colleagues, and get my hands dirty in many math courses.

Last week one of my “regulars”, who often leaves class early for sports, came to the lab for Algebra 2 help.  He missed out on a adding/subtracting rational expressions lesson, but had the notesheets handy.  We wrote the first problem on the board:

Nothing too fancy. But in math lab, I don’t know the students as well as my own, so I need to do a quick check for some background before diving into new material. A spot check for understanding of fraction operations was in order:

The student approached the board confidently and “added” the fractions:

…sigh…. sometimes there isn’t enough coffee in the world. But all is not lost, and after a stare-down, the student recognized he had acted too quickly, and completed the problem correctly. This led to another problem. This time, I asked the student to just tell what me what the common denominator would be:

Without hesitation, the student knew the correct denominator to be 24. But why is it 24? The student could not defend his answer, but was absolutely sure 24 was the LCD. On the one hand, I am happy that the student has achived enough fluency with his number sense to confidently find the denominator. But, on the other hand, has lack of a process is going to hurt us now when we try to apply LCD’s to algebraic expressions.

When I start my lesson on adding rational expressions, I hand out index cards to every student.  I give students 2 minutes to repond to the following prompt:

How do you find a least common denominator? Provide directions for finding an LCD to somebody who does not know how to find one.

I collect all of the cards, shuffle them, and choose a few randomly to share under the document camera.  We will discuss the validity of the explanations, and use parts of the explanations to come up with a class-wide definition of an LCD.  Here is what you can expect to get back on the cards:

• Some students will recognize that factors play a role, but won’t recognize that the powers of the factors are imporant.
• Many students will attempt to use an example as their definition.  This allows for a discussion of a mathematical definition.  Is one example helpful in establishing a rule?  How about 2 examples?  How are examples helpful, if the reader does not know how to find an LCD?
• Some students will provide a hybrid of the last two bullets.
• If a student does provide a suitable definition, it’s time for you to play dumb.  Let the class assess the language and verify that the definition is, or is not, suitable.  In one class, I “planted” a working definition in with the student cards to see if they could identify a working definition.

The Least Common Denominator of two or more fractions is the product of the factors of all denominators, raised to the highest power with which they appear in any denominator.

HOW DO EARLIER MATH EXPERIENCES PROVIDE A SUITABLE BACKGROUND FOR RATIONAL EXPRESSIONS IN ALGEBRA?

I am curious how math teachers approach Least Common Denominators in earlier grades, and how these approaches translate to algebra success.  Here’s how a few online math sites approach LCD’s.

First up, mathisfun (search for “Least Common Denominator):

This is the approach I suspect many teachers take to help students find LCD’s.  It works for manageable denominators, but becomes cumbersome when we have 3 or more denominators to consider, and certainly is not helpful in our algebra world.  Also, if you get too confused, you can use the “Least Common Multiple Tool” this website provides.  I suppose it’s not an inappropriate method, but a more algebra-friendly process should eventually develop.

Next up, Everyday Mathematics at Home website:

The good news – we have a formal definition!

The bad news – you have to know what an LCM is to use it.

This is a more formal version of the Mathisfun example.  We could adapt it for use in algebra, but again, a definition of LCM is required here.

Khan Academy starts by using lists of multiples and provides and example with a trio of numbers for which we want the LCM:

The factor trees and the color verification that all 6, 15, and 10 are all factors of 30 is nice, but this example conveniently leaves out any scenario where a number is a factor multiple times, and this is the only example given.

Finally, let’s check out how PurpleMath tackles LCM’s, with a non-intuitive example:

Now we are getting someplace.  Not only does this method stress the importance of factors, it shows the importance of include all powers of those factors.  And I could transfer this method easily to algebra class!

Have any insihgts into teaching LCD’s, either for a fractions unit, or in algebra?  Would enjoy hearing ideas, feedback and reflections!