• ### PART I

PLACE VALUE, ARITHMETIC MODELS & ARITHMETIC ALGORITHMS

MENTAL MATH

• ### PART III

PRIMES & DIVISIBILITY

• ### PART IV

FRACTION ARITHMETIC

• ### PART V

WORD PROBLEMS & MODEL DRAWING

# Part2i: Mental Math Introduction

One of my favorite movies is Stand and Deliver, in which Edward James Olmos portrays legendary teacher Jaime Escalante. In one of the more memorable scenes of the movie, Escalante demonstrates the so-called ‘fingers method’ for multiplying by 9(www.youtube.com/watch?v=-WOgLltWhgg). Most teachers with whom I have worked and many of my students are familiar with this technique, yet few if any can explain why it works. The fingers method is just one of scores of similar computational ‘tricks’; in fact you can find roughly 7760 videos illustrating such tricks on YouTube by simply typing ‘math tricks’ into the search engine. I didn’t have time to survey each of these, but I’m guessing few if any contain an actual mathematical explanation for why the trick works. But there are no tricks in math; a logical explanation can (and should) always be given. And at the elementary level such explanations need not involve any fancy mathematics; they can (and should) always be made grade appropriate.

One might at this point wonder ‘Why does an explanation matter’? Isn’t it enough for students to have a nice, easy, way to figure out how to mentally multiply by 9? To examine this issue, let’s consider the following hypothetical situation. Suppose that Teacher A teachers the fingers method to her class; and Teacher B teaches her class the mathematical reasoning that makes the fingers method work, i.e., the distributive property. What types of problems can these sets of students now solve? Teacher A’s students can use the fingers method to solve any multiplication problem whose factors are 9 and a single-digit number . But that is all. This method cannot be used on a problem such as  for obvious reasons (we don’t have 12 fingers).

On the other hand, Teacher B’s students can not only mentally solve any reasonable multiplication problem in which 9 is a factor (including the problems Teacher A’s students can solve, but also problems like  and ), but these students can also use the distributive property to solve problems like , and , and even problems like  and . Moreover, these students will gain an appreciation of the power of the distributive property, and will be better prepared for algebraic manipulations that require its use.

In this section we focus on Mental Math, which is exactly what it sounds like; Mental Math consists of techniques for mentally performing arithmetic computations involving whole numbers, decimals, and fractions.   I am careful to use the word ‘technique’ as opposed to ‘trick’; all techniques are based on properties, models, and interpretations of arithmetic that were illustrated in Part 1. The last clip in the section (Clip 2 ix) contains solutions to Mental Math exercises I gave to a group of in-service teachers; the problems themselves are contained in an attachment if you want to practice what you learned in the clips. Good luck, and have fun!

Notes:

1. Clips 2 ii through 2 vii focus on individual strategies, while clip 2 viii is a much longer, classroom discussions of all the strategies at once.
2. When re-watching clip 2 ix, I noticed that I repeatedly used very poor language when discussing decimal numbers. In the future when I am able to add a section on decimals, I will make a point to mention that teachers should be very careful with the language they use. For example, when saying the number 4.27 aloud, one should not say ‘Four point two seven’ (a mistake I made several times in the clip), but instead, one should say ‘Four and twenty-seven hundredths’.