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Section 28.5 Going Deeper: Mental Arithmetic

For many students, mental arithmetic usually feels quite complicated and can be extremely frustrating. One of the big challenges for mental arithmetic is that most people try to do mental arithmetic the way they do arithmetic on paper. Unfortunately, this is both extremely inefficient and unnecessarily difficult. The primary reason has to do with how the paper method relies on being able to write down a digit to remember it later, but our brains simply cannot do that.

To get a sense of just how central that memory is, we're going to work through adding three three-digit numbers together, and we're going to put a box around every number that appears in the process:

\begin{equation*} \begin{array}{r} 387 \\ 236 \\ + 523 \\ \hline \end{array} \end{equation*}

  • Ones digit: \(\boxed{7}\) plus \(\boxed{6}\) is \(\boxed{13}\text{,}\) plus \(\boxed{3}\) more is \(\boxed{16}\text{.}\) Write the \(\boxed{6}\) below and carry the \(\boxed{1}\text{.}\)

  • Tens digit: \(\boxed{1}\) plus \(\boxed{8}\) is \(\boxed{9}\text{,}\) plus \(\boxed{3}\) more is \(\boxed{12}\text{,}\) plus \(\boxed{2}\) more is \(\boxed{14}\text{.}\) Write the \(\boxed{4}\) below and carry the \(\boxed{1}\text{.}\)

  • Hundreds digit: \(\boxed{1}\) plus \(\boxed{3}\) is \(\boxed{4}\text{,}\) plus \(\boxed{2}\) more is \(\boxed{6}\text{,}\) plus \(\boxed{5}\) more is \(\boxed{11}\text{.}\) Write down these digits to get the final result.

Let's make a list of all the digits that appeared in the process, keeping track of the order in which they appeared:

\begin{equation*} \boxed{7} \, \boxed{6} \, \boxed{13} \, \boxed{3} \, \boxed{16} \, \boxed{6} \, \boxed{1} \, \boxed{1} \, \boxed{8} \, \boxed{9} \, \boxed{3} \, \boxed{12} \, \boxed{2}\, \boxed{14} \, \boxed{4} \, \boxed{1} \, \boxed{1} \, \boxed{3} \, \boxed{4} \, \boxed{2} \, \boxed{6} \, \boxed{5} \, \boxed{11} \end{equation*}

When you look at this string of numbers, the critical observation to make is that the actual answer to the calculation does not appear anywhere. In fact, when you look at the numbers, you may have a hard time even finding the digits that make up the answer. We'll indicate those values with an arrow:

\begin{equation*} \boxed{7} \, \boxed{6} \, \boxed{13} \, \boxed{3} \, \boxed{16} \, \underset{\big\uparrow}{\boxed{6}} \, \boxed{1} \, \boxed{1} \, \boxed{8} \, \boxed{9} \, \boxed{3} \, \boxed{12} \, \boxed{2}\, \boxed{14} \, \underset{\big\uparrow}{\boxed{4}} \, \boxed{1} \, \boxed{1} \, \boxed{3} \, \boxed{4} \, \boxed{2} \, \boxed{6} \, \boxed{5} \, \underset{\big\uparrow}{\boxed{11}} \end{equation*}

Now that they've been marked, notice how many numbers appear in between them. These digits are numbers that you're trying to hold in memory while all the other numbers are going through your head. Between the 6 for the ones digit and the 11 for the hundreds and thousands digit in the last step, there are 16 numbers in between, and one of those other numbers is one you were supposed to have memorized.

When following this algorithm on paper, our brains are allowed to forget those digits because they're written down. But when we do this mentally, we have to try to keep those numbers in memory while other digits are being processed. While it helps to have reached a level of automaticity so that you're less consciously doing those computations, it's still quite challenging for most people.

The standard addition algorithm uses a digit manipulation scheme. What this means is that we're looking at each digit as its own object. In the calculation, the number 387 is never used as a number. It's broken down into three separate pieces:3, 8, and 7. This causes extra strain on our brains when handing it because it requires more brain power to think about three separate objects rather than thinking about one.

The big transition that makes mental arithmetic easier is to reorganize the calculation so that we don't have to remember numbers for as long. The representations of arithmetic on the number that we developed a few sections ago are a big towards that end. Here is how we're going to translate the process:

\begin{equation*} \begin{array}{r} 387 \\ 236 \\ + 523 \\ \hline \end{array} \quad \longrightarrow \quad \begin{array}{l} 387 \\ + 200 + 30 + 6 \\ + 500 + 20 + 3 = \end{array} \end{equation*}

Here is how we're going to work through it:

  • Add the first two numbers: \(\boxed{387}\) plus \(\boxed{200}\) is \(\boxed{587}\text{,}\) plus \(\boxed{30}\) more is \(\boxed{617}\text{,}\) plus \(\boxed{6}\) more is \(\boxed{623}\text{.}\)

  • Add the next number: \(\boxed{623}\) plus \(\boxed{500}\) is \(\boxed{1123}\text{,}\) plus \(\boxed{20}\) more is \(\boxed{1143}\text{,}\) plus \(\boxed{3}\) more is \(\boxed{1146}\text{.}\)

This is how those numbers look written in order:

\begin{equation*} \boxed{387} \, \boxed{200} \, \boxed{587} \, \boxed{30} \, \boxed{617} \, \boxed{6} \, \boxed{623} \, \boxed{623} \, \boxed{500} \, \boxed{1123} \, \boxed{20} \, \boxed{1143} \, \boxed{3} \, \boxed{1146} \end{equation*}

Let's make some observations about this:

  • The total number of boxed values is significantly lower. There were 23 in the first method and only 14 in the second method.

  • However, the numbers that we were working with are also significantly larger. In the first method, we only needed to be comfortable with small addition calculations, whereas the second method requires a level of fluency with adding in different place values.

  • The new approach does not require any long term memory, and the final number is the complete final answer. In fact, the process is basically just keeping a running total as you work your way through, so that you only really have to remember keep three numbers in your head at any time (the previous total, the number you're adding, and the new total).

As with most things mathematical, this should not be seen as a "rule" for how you're "supposed to" do mental arithmetic. It turns out that people who are good at mental arithmetic employ a number of different techniques based on patterns that they identify as being familiar. It's not necessary to always go in the same order every time, especially if there are patterns that make more intuitive sense.

For example, in this calculation, you might notice that \(300 + 200 + 500 = 1000\text{.}\) This is a number that you can put into memory (with a little bit of practice) because it's a "nice" number. And then you would only need to add three two-digit numbers together instead. You might also notice that \(387 + 3 = 390\text{,}\) which also brings you to a nice number. Then you can add in the hundreds and tens digits, leaving the final 6 for the very last step.

You could also use other groupings if they make sense. For example, you might see

\begin{equation*} 387 + 236 = 387 + 13 + 223 = 400 + 223 = 523 \end{equation*}

and do that faster than you would do

\begin{equation*} 387 + 200 + 30 + 6 = 587 + 30 + 6 = 617 + 6 = 623. \end{equation*}

The main point is that mathematics favors those who are flexible in their thinking instead of being rigid. Throughout this book, we've emphasized just how restraining a rule-based approach to mathematics can be, and this thread follows all the way back to the ways that you've learned your basic arithmetic. Hopefully, as you've been working your way through this book, you've started to move away from those things and have started to build a deeper understanding of mathematics.