By John SanGiovanni (@JohnSanGiovanni), coauthor of the *Mastering the Basic Math Facts* series

Stop me if you’ve heard this one before. Three students walk into a math classroom. They confront 709 – 340. Krista uses a number line to count back with three jumps of 100, a jump of 50, and another jump of 19. She then adds her jumps. Damian solves with paper and pencil. Oscar counts up mentally from 340 to 640 (300) and then 640 to 709 (69 more) to compose a difference of 369. So which student selected and used tools strategically?

The answer depends on our interpretation of tools, our expectations for using tools, and the mathematical maturity of our students. So what constitutes a tool? (Insert joke here!) Clearly, protractors, rulers, graphing calculators, and manipulatives like color tiles qualify. Paper and pencils do, too. Simply, a tool is anything that aids in accomplishing a task (including Krista’s number line).

What *is* unique about Practice #5 is the idea that these tools are used strategically. In many situations, paper and pencil are inefficient and using them is not strategic. We must develop the notion that mental computations are possible, reliable, and often more efficient. Oscar shows mental computation at its finest. Consider Krista and Damian. Which pathway is strategic? It depends if they are a third grader, a fifth grader, or a ninth grader.

Using a calculator should also be a strategic decision. It should be a tool that provides access, simplifies the task, or confirms accuracy. It doesn’t make sense for a fifth grader to use a calculator for 8 + 13. However, it may make perfect sense for a first grader to confirm the sum that way. Consider 8.1258 + 12.14920. Is this a mental math activity? Does it call for paper and pencil? Does it make sense to use a calculator? Better yet, do our students have opportunities to discuss when they might use either of these tools? Do our students have the freedom to use or not to use these tools as *they* deem necessary?

We must also recognize that tools do not produce understanding, problem solving, and solutions. These come from the individual. I cannot assume that providing a student with a protractor ensures that they will find angle sums with accuracy. Similarly, a graphing calculator doesn’t consider user error or misconception when graphing a linear equation.

User error (including broken buttons) can occur when using basic calculators. Yet when something goes wrong, we pause. We consider. We adjust. We try the tool again. How does our thinking factor into using tools? What questions do we want our students to ask when selecting and using tools? To me, these key questions include:

- “Do I need a tool?”
- “What tool is the best to use?”
- “How does it work?”
- “Do the results align with what I was expecting?” or “Do the results make sense?”

Concepts must be developed with the tool, even reinforced by the tool. The use of the tool itself should support reasoning rather than mere procedure. In other words, students must understand angles prior to using a protractor but also reinforce this understanding while using the protractor. Using a protractor is more than “lining it up the right way.” Understanding enables them to be proficient in diverse situations and even with diverse protractors. Consider these four examples:

They all measure angles but each does so in a different way. What understanding must a student have to use each of these effectively?

Understanding helps us realize accuracy. Consider 8.1258 + 12.14920 from above. If I use a calculator, I should know that my sum will be in the neighborhood of 20. I need to reconsider if my calculator result is dramatically different. The idea of understanding holds true for other tools including paper/pencil. It transcends grade level. For example, if I determine that my slope is negative and my line rises from left to right, then something is awry.

We play a critical role in the development of strategic use of tools. First, we make tools available to our students. We encourage their strategic use. We model using them. We don’t admonish students who choose to use them. Instead, we ask them to share their reasoning for using a tool. We identify how tools connect to the ideas of mathematics. We make varieties of tools available for a situation and provide variety in the tools themselves when possible (i.e. different protractors).

Most importantly, we constantly develop the metacognitive process of tool selection and use. This development is centered on the bulleted questions above. We lift up the thinking behind the tool as well as the procedure for using the tool. We require our students to either predict what their finding might be prior to using the tool. Or, we require a reflection on the results and if they make sense.

Tools *are* meant to make sense of mathematics and the world around us. They are meant to improve efficiency and support accuracy. They are a part of modern life—as is our responsibility to understand them, use them strategically, and develop these proficiencies in our students.

John SanGiovanni is the coauthor of the best-selling books *Mastering the Basic Math Facts in Addition and Subtraction* and *Mastering the Basic Math Facts in Multiplication and Division*. He is also the coauthor of *Putting the Practices Into Action*, which examines the Standards for Mathematical Practice and unpacks their power for developing deep mathematical understanding in students. Follow him on Twitter @JohnSanGiovanni.

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Next week, June Mark, Paul Goldenberg, and Jane Kang explore SMP #6: Attend to precision. Click here to see the entire SMP series lineup and sign up for email alerts when each new blog is posted.

SteveHThese are not new ideas. When I was in grade school long before calculators, we didn't just learn standard algorithms by rote. We had partial sums and we knew exactly how borrowing worked. We learned to do mental math and the value of estimating. Long division might require you to figure out how many 23's went into 163 in your head. Everyone learned how to do n*20 + n*3 in their heads. Nobody raised that up to some sort of pinnacle of "strategic" math understanding and application. We all were taught counting up as a mental math option. We also learned to linearly interpolate numbers from tables in our head – a very good skill to have that is completely missing now. It's amazing how modern educators have redefined and vilified traditional math. The standard (efficient versions of) arithmetic procedures were important tools because we did not have calculators. So now we have calculators and the importance of paper and pencil tools for basic arithmetic is less important. However, this comparative analysis goes away as soon as you get past fourth grade.

What's missing these days is mastery of the tools and the understanding that goes with them. This has to happen before any strategic application can be done. Too many educators are interested only in the explanation rather than the result, and they extrapolate from simple math to more complex cases. If a student tries to do count up and gets it wrong, but adds in words that explains a counting up process, teachers are thrilled. It was just a simple mistake, like a typo. Mastery is seen as just efficiency.

This analysis falls apart when you get to a more complex tool like algebra. There are simple "dumb" mistakes and there are mistakes that show a lack of understanding of how the "tool" works or how to apply it. All math teachers have seen this, and more often than not, students have understanding problems and not "typo" problems. More importantly, these are understanding issues that cannot be solved or proven with words or even proofs. When I tutor a student, I can see they "understand" and they can use words to show that. However, there are too many subtle variations that test that understanding – variations that simple words cannot explain. We are no longer in the words + typo domain. We are in the area where understanding is shown only by proving you can do the problems correctly. I tell my students that they really, really have to go home and do the problems themselves. Invariably, they come back with questions that show they really didn't fully understand.

This understanding is only accomplished by mastering nightly homework sets and having a process that gets that done. However, curricula like Everyday Math do not do that. They tell teachers to "trust the spiral." Full inclusion in K-6 is a fundamental flaw because nobody is pushing and expecting mastery "at any one point in time", according to Everyday Math. CCSS does not fix this fundamental flaw. It talks about fluency, but it is vague and minimal at best, and the highest level on tests like PARCC only mean that one is likely to pass a course of college algebra. This highest level starts in Kindergarten. I even had to ensure mastery of basic skills at home for my "math brain" son. Schools don't like tracking and ensuring mastery so it gets hidden at home. Schools don't ask the parents of their best math students what they had to do at home. I got to calculus in high school with no help at all from my parents. Now, we parents get notes telling us to practice "math facts" at home. No amount of vague SMP words will fix this.

Bryan PI’m not sure mental calculations are more efficient that paper and pencil in all cases. I would much rather have paper and pencil to subtract 13725 – 2756 rather than rely on my working memory. I find it a bit short-sighted that you would go to such lengths to say that paper and pencil is just ‘not good enough’ anymore. You are basically saying “you must calculate my way or you are incorrect.” This again seems odd, as most educationalists believe in ‘multiple strategies’ and using paper and pencil along with a standard algorithm is a very reliable strategy to find the solution of a given problem. So which is it? Multiple strategies, in which case paper and pencil along with the algorithm has its proper place, or a one-size fits all, in which case the best strategy to alleviate cognitive overload is to teach the paper and pencil strategy of the standard algorithm?