Naked Math is the Root of all Evil

algebra2 NakedMath.jpg

Perfect example of a "naked math" problem:  absolutely NO context for understanding

I spent many, many years in school learning mathematics and training to be first a teacher, then a teacher-educator.  Out of that lengthy experience, a few key things remain etched in my memory. One of them is the phrase "naked math" that I coined after hearing prolific mathematics education researcher and wonderful educator, Constance Kamii discuss 'naked numbers' in addition and subtraction problems. I use the term naked math to describe a problem (much like the one shown above) that does not allow for analysis or evaluation because there is no scenario or setting from which it arose.

In my personalized definition, naked math is math without context, without a frame of reference for understanding, and without sufficient details to allow one to form a mental picture of the process, model, or procedure required. For the record, I am opposed to naked math.  I think it should be avoided as much as possible. It should especially be avoided when learners are seeing a concept or procedure for the first time.  In fact, the desire to ensure an appropriate context within which to understand mathematics is why I chose "No Naked Math!" as one of my company's slogans (it's also one of my personal mantras, along with other classics such as "It's better to ask forgiveness than permission").  

One of the primary reasons I'm not in favor of most standardized testing is because they are filled with naked math problems.  As a mathematics educator, I often teach people "what to do when you don't know what to do" (aka problem solving) and, when I do, I use Polya's problem solving process.  It's simple and it works. The last step in Polya's process asks the learner to "reflect back" on the solution and to ask the very important question, "Does my answer make sense?"  Please note that with a naked math problem, this is virtually impossible to determine. 

Another element permanently etched into my memory from my education and experience is the term "numeracy" to describe the kind of fluency with numeric thinking and problem solving which we all desire to possess and impart to the next generation.  [Also the term "innumeracy" to describe the kind of numeric illiteracy that afflicts too many of us, even after an educational experience. For more on this, check out John Paulos great book by the same title.]

In the UK, there's a great site that describes their National Numeracy initiative.  I love their description (below) which directly challenges the "math gene" myth that is prevalent here in the US.

We see changing people’s attitudes as a vital element. Maths is not something that you either simply can or cannot do. Maths is challenging and demands persistence in order to progress, whatever level you are working at. In order to achieve, learners need to persevere to overcome the challenges that are a part of learning maths for everyone; and they will only do this if they believe that it is possible and important to achieve in maths.

To me, one of the easiest and most logical ways to convince someone that it is possible to do math and important to achieve in math is to place it in a context with which they are already familiar.  By "clothing" the math in a context with which the leaner is familiar, it allows application of all types of problem solving strategies (often referred to as 'common sense'), not just those restricted to classroom mathematics.  In my experience, learners get much more confident in their own abilities after successfully working on a problem that they can think about and try to figure out (rather than just spending time trying vainly to recall an obscure math procedure or rotely-memorized rule). 

To clarify a bit more, allow me to give a naked math example along with an equivalent rich mathematical task to illustrate the vast differences between the two.

Naked Math Example

“Naked math” example - math without context.

Note that you have no idea what these equations are modeling, what the variables are representing, or where in the real world this system of equations might arise. Hence, a perfect "naked math" example.

Rich Mathematical Task (RMT)

I first learned the phrase "rich mathematical task" from NCTM.  It refers, in my opinion, to the antithesis of a naked math problem.  Here's an example of a great RMT from a wonderful article by Joe Garofalo, my dissertation advisor, & Christine Trinter.

Softball RMT

Potential questions:

1.  What is the maximum height reached by the softball?

2.  Will this contact result in a home run?

Notice that in the RMT about the softball problem, students are required to create a system of trigonometric equations and then use them to answer the questions that arise from the familiar context of a softball game.  Also notice that the system of equations is equal in difficulty (if not more than) the naked math example:

Softball equations

Most importantly, though, notice how now, in a context, the coefficients all have meaning:  the 104 we know comes from the fact that the initial velocity of the ball is 104 f/s, the 20 we know comes from the angle of elevation being 20 degrees, the 3 comes from the fact that the player struck the ball 3 feet above the ground, etc.  The math has meaning because of the context, and, because of this, learners can think about it much more deeply and intelligently. Not to mention, they are also often more motivated to find the answer!

This application of math to model situations, create simulations, and solve problems is one of the most important abilities our students can learn.  Learning this vital skill does not happen in a semester, cannot be tested with a single multiple-choice assessment, and is not a typical result in our K-12 educational system.  Other nations struggle to create a system that results in students with this ability as well.  In the National Numeracy site, the phrase "mathematical journey" is used to describe the life-long process required.  I really like this phrase and the corresponding imagery.  Below is a description & a graphic of the "mathematical journey" from the UK's site:

A firm foundation in understanding numbers, the number system and the ways that numbers can be combined and used is essential to numeracy confidence and competence, but on its own it does not provide people with functional skills in their daily lives.

For everyone needs also to be able to apply the skills and knowledge in all the varied contexts of their daily lives. They need to be able to solve problems, interpret information and make informed choices.

When they are calculating, they need to use the most appropriate and efficient methods and know if the answer is reasonable. When trying to solve a problem in a particular context, they need to be logical and systematic, ready to persevere but also flexible. The following diagram illustrates these essential components - all important aspects of being numerate and having this life skill.

the-essentials-of-numeracy.png

Now that you know that naked math is the root of all evil in mathematics education, I hope you will strive to include context as much as possible, especially when introducing new topics.  Once students have been exposed to sufficient potential real-life models for any particular mathematics content, then you may throw in some naked math if you must, just for the sake of practice, but not until, or you risk your students numeracy.  

As always, I'd love to hear from you: link.

References:

The Case Against Math post (& where the problem graphic came from)

Collection of articles by Constance Kamii

Constance Kamii's page

Mathematics in Context (MIC) philosophy statement

Fabulous video of Sir Ken Robinson on "Educating the Heart & Mind"

NCTM (wonderful resource for rich mathematical tasks with meaning)

Article by Joe Garofalo & Christine Trinter (where RMT came from)

The UK's National Numeracy site

The Mathematical Journey described

Innumeracy by John Allen Paulos

Not related to naked math, but very cool, my colleague Michael Naylor's work in Naked Geometry


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The Solver Blog

Author:  Dr. Diana S. Perdue

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