The "Wolfram Alpha" Era

If any of you have ever read Principles to Actions (2014) by the National Council of Teachers of Mathematics, you are familiar with the 8 Mathematics Teaching Practices that are addressed in the chapter on Effective Teaching and Learning.  If you haven't read Principles to Actions (2014), seriously you should--I have read it at least 3 times, and I still get something out of it every time I read it!! 

Within the 8 Mathematics Teaching Practices, there is one specifically that I have been thinking about: Build Procedural Fluency from Conceptual Understanding (NCTM, 2014).  Just to be clear, procedural fluency, as defined in Adding it Up (2001) by the National Research Council, is "knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently (pg. 115)" and conceptual understanding is "comprehension of mathematics concepts, operations, and relations (pg. 5)."  

The reason Ive been thinking about this teaching practice is because of a fabulous website called Wolfram Alpha.  It is a computational knowledge engine.  So, pretty much anything that has to do with something numerical, it can tell you something.  For example, if I search my name, Taylor, as a given name, this is what came up: 

I know... it's pretty awesome!!! You could search weather, time, history, health, geography, science, and that is just a start! It can even solve difficult equations for you.  

So, here is my question: In a world where Google, Wolfram Alpha, and so many other tools like this exist, what does it really mean to be procedurally fluent? 

I remember being in high school from 2004-2008 and Google existed, but it was not as widely used or popular as it is today. It is only lunchtime, and I have already googled something 5 times today.  I remember teachers telling me it was important to memorize formulas, trig identities, etc. because in the "real world" there wouldn't be someone to give me the formula I needed to solve a problem.  Well, guess what... now there is! 

The question still remains, where is the line between making them memorize a formula and compute by hand, and teaching them to use the tools they have in an appropriate way? 

Here are my initial thoughts on this question, and I would appreciate any and all feedback.

First, I think we should teach them to solve problems on their own AND how to use tools like Wolfram Alpha to help them.  Mathematics Education today--what I have currently named "The Wolfram Alpha Era"--we have to adapt.  We have to teach our students what tools are out there for them to use if they are going to be competitive in today's workforce. However, we cannot completely change our ways until the way that our state/country assesses our students also changes.  We cannot rely solely on Google and Wolfram Alpha when they cannot use those tools on standardized assessments or in colleges.  So I guess my point is, we must teach them to be adaptable. If you refer to the earlier definition of procedural fluency, you will notice a section that says, "skill in performing them flexibly, accurately, and efficiently"(National Research Council, 2001, p. 115). All that requires adaptability in using the most appropriate tool in the given situation, even if that tool is Wolfram Alpha.  

Second, when we use tools like Wolfram Alpha and Google, we can teach our students how to ask the right question. When you use Wolfram Alpha, just by typing in a very direct question, doesn't necessarily mean you will get a direct answer.  In class the other day, I typed in, "What is the formula for the first n perfect squares?".  Wolfram Alpha didn't understand my question.  I tried again: "Summation of perfect squares".  Still no result.  There were a few other attempts, and then I tried "sigma x^2 from 1 to n", and it gave me the formula I wanted.  My point is--I asked a very direct question. I thought it was clear. The search engine however, didn't understand until I asked the right question.  We can use these tools to teach our students to reword their questions and to be more specific in their thinking.   

Finally, we can use these tools to motivate them mathematically.  Just because you type a problem into Wolfram Alpha, doesn't mean it will give you the answer in the form you typically use.  You might get an equation in point-slope form when you typically use slope-intercept.  You might get the exact solution to an equation, when your students are more comfortable with the decimal approximation.  Upon receiving this unfamiliar form, we can motivate them to explore why they are mathematically equivalent and what mathematics would they do differently to get the unfamiliar form.  

I would appreciate all thoughts/comments that you have! :) 

References: 
National Research Council. (2001). Adding It Up: Helping children learn mathematics. Washington, DC: The National Academies Press.

National Council of Teachers of Mathematics. (2014). Principles to Actions: Ensuring mathematical success for all. Reston, VA: National Council of Teachers of Mathematics.