Negative exponents

Using powers of two for explaining the concept of negative exponents

Negative numbers confuse many students. Usually students tend to struggle with almost everything related to negative numbers. From the very concept of using the (-) sign to refer to conventional spatial directions (left, down, back), to the different rules for adding, subtracting, or multiplying positive and negative numbers; there are plenty of instances where the (-) sign is overlooked, or misinterpreted, resulting in a wrong answer. Given that a high number of wrong answers produce a low score, it is only natural for many students to react defensively whenever a new concept involving negative numbers shows up in their radar. This is the case for negative exponents. Remember that the first definition of exponent (positive) most students are introduced to is: “the number of times you multiply a number times itself.” When one tries to apply this definition to the case of negative exponents it does not make sense because, how do you multiply a number times itself “minus once,” or “minus twice”? One possible answer is that, when it comes to negative exponents, you do not multiply but you divide instead. Even with that interpretation the rule still needs some adjustment because, if you start with a number a, and you divide it by itself once, you get 1 as a result, so you would think a^-1 was 1 but that would be wrong because 1 = a⁰, whereas a^-1 = 1/a.
In order to avoid this kind of confusion, I follow another approach: By asking very simple questions I help the student build a list of powers of 2 from 2 to 1024, each related to their corresponding positive exponent. Then we notice the patterns how the numbers change, and how the exponents change. Then we move backwards all the way to the beginning of the list, and beyond, continuing into fractions, on one column, and into negative exponents, on the other column. In this way it becomes crystal clear to the student how the exact same pattern correlates the negative exponents to the fractions, and even the general formula a^-n = 1/aⁿ becomes apparent. I have successfully used this method many, many times, and so far it has worked wonderfully, clearing away all confusion and anxiety students of all ages had around negative exponents. The key to this approach is being very thorough, and asking the right questions, in the right way, at the right time.

Math is not English

The order of operations messes with our reading habits.

The mathematical order of operations seems to be a source of confusion for some students, sometimes even frustration. For example, when presented with the expression
3 + 4(x-1)
some students ask: “Why can’t we just start by adding 3 + 4, and then multiplying 7 times (x-1)?”
My short answer is: “Because math is not English.” Then I ask: “Is there any parenthesis around the 3 + 4 sum?” When they say “no” I continue: “Then the parenthesis that is there right after the 4 claims that 4 for itself, for multiplication purposes. It will not let the 4 run away with the 3, oh no sir, no way! The multiplication operation has title to that 4, and to that (x-1) as well, and it does not care about the 3 the slightest bit. The addition operation holds a lesser priority than multiplication does, so it has to wait for its turn.”
I explain the PEMDAS rules using action verbs commonly applied to human situations, thus making the math symbols play the role of active, independent characters with human-like behaviors. This type of explanation makes my students understand the mathematical structure of the expression at hand but still some seem puzzled, or surprised, or even bothered by the fact that the applicable sequence of operation does not necessarily follow the simple left-to-right order. So in those cases I proceed with the following explanation:
There is a crucial difference between the way we read math, and the way we read English. This is very important. We always read English from left to right. Such a simple, linear, unidirectional way does not do it for math. It does not work. Reading math from left to right only, is insufficient, and inadequate. In math we have to read formulas and expressions not just from left to right but from right to left; from the top down; from the bottom up; from the inside out; from the outside in; and even around in circles. In short, every which way, else we run the risk of missing essential information about the structure of the thing. Reading math is not reading. Reading math is much more similar to what the eyes of a helicopter pilot do when they are flying over a mountain terrain, looking for a spot to safely land the helicopter. You look at everything, everywhere.
Written language mimics spoken language, going along with the flow of the story. English is perfect for telling stories. Math describes structures. It has an altogether different goal, so it cannot work the same way English does. Mathematical expressions do not resemble stories nearly enough the way they resemble gizmos, appliances, devices, or cars, objects made out of parts. The parts are connected to each other in a very specific way. Each part has its own function, and its own place within the whole thing. So, really, reading math from left to right only, makes as much sense as trying to “read a car” from left to right only.

Finding Problems That Motivate Students

Time units may help with multiplication practice

According to my experience each child is a unique learner. Generally speaking, students are more motivated to solve a particular problem when the problem relates to something they find real, meaningful, or important. However, different children usually assign different degrees of importance to the same thing. So it is always helpful to find a topic that holds a student’s interest, and that can be easily connected to math.
For example, let’s say you want to help a student who is struggling with multiplication tables. You can start a session by asking how many hours are there in a day, how many minutes in an hour, days in a week, and so on and so forth. Most third graders will know the answers off the top of their head. Then you proceed to ask how many minutes are there in a day, how many hours in a week, and so on and so forth. Children will realize these are multiplication problems but each child will react with a different degree of enthusiasm –or apathy– to find the answer.
Students who somehow have a strong connection to time units will diligently work out these multiplication problems all the way down to the number of seconds in a week. With a little bit of help they will continue working until they get all the numbers right, even if they struggle with the multiplications tables all along. They do this because they want to know the answers. In their mind, these problems are real, not just an empty drill.
Some other kids do not care at all how many seconds are there in a day, so for them this particular set of problems will not be very motivating. You will have to find a different set of problems for them.
Even when students understand the concept of multiplication, and the basic rules to multiply numbers, they may feel that some questions do not justify doing all the work necessary to find the answer. It depends on how real or important the questions seem to them, because that is what makes them want to find the answers.

Dealing with the difficulty of memorizing products of digits higher than five

Advantages of using Mayan and Roman numerals

When it comes to memorizing the multiplication tables, each child has his or her own pace. Some children find it easier than others to memorize the lists of numerical facts that make up the multiplication tables. Others need longer practice periods, maybe with the help of flash cards. Rote memorization is not everybody’s best act. Some children find these dry memorization exercises burdensome.
Here I offer a suggestion to help third and fourth graders who are not into rote memorization, to gain a better grasp on multiplicative manipulation of the higher digits.
The main idea is using the distributive property of multiplication over addition to calculate the result of a product of two “big” digits by breaking one of the “big” factors into smaller ones, doing two smaller multiplications, and adding up the partial results at the end.
The essential keys for this approach to be successful are:
First of all, do not even mention the phrase “distributive property.” That’s a big no-no. Do not do it. Show by example only. At this stage children do not need to know there is such a thing as a distributive property. It would be a waste of time. Avoid the confusion. Just do it. Show them how it works but do not try to explain why. The best way to understand why it works is for them to see how it works, period. Do not say aloud any abstract names like “property,” much less “distributive.” Stick to the numbers.
Second, use a standard “breaking scheme.” The number five is an excellent stepping-stone in this process. It is very natural for a variety of reasons. Mainly, because five is half of ten, the base of our numerical system, plus we have five fingers in each hand, so it is very easy to break any higher digit as the sum of five plus a lower digit. Children accept this fact very easily. Furthermore, the multiplication table of five is one of the easiest to remember.
Third, –and this is very important– spend enough time (at least half-an-hour) in a preparation period showing them, or reviewing with them, how to write numbers using Roman numerals and Mayan numerals. Do this before getting into any multiplication practice. This specific type of preparation has a dual purpose. On the one hand, it lowers their anxiety level. You have to understand they are under pressure. Their parents are at least concerned, maybe even worried. That is why they hired a private tutor in the first place. The child knows he or she is not doing great in the class when it comes to memorizing the multiplication tables. Some children may be even beginning to have some dents in their self-esteem, thinking that perhaps there is something wrong with them, or that they are not good at math, or whatever. The main idea in their head at this time is “multiplication is hard.” So when you –the expert– come along and start working with them doing lists of Roman and Mayan numerals, they go “Oh! This is really not that hard. This is easy.” Some children actually have fun with these numerals. Some prefer Roman numerals, while some prefer Mayan ones. The main point is now they are relaxed, at ease, and working with something they understand much better than the monolithic multiplication tables. That is the first goal of this preparation. The second purpose is for them to realize, or remember, or reinforce the idea of just how natural is the use of the number five as a breaking point, or a stepping-stone. Both Roman and Mayan numerals make heavy use of the number five, and of multiples of five, in this fashion. They consistently apply the principle of expressing higher digits as sums of five plus a lower digit. After doing this work with Roman and Mayan numerals, children are so much more receptive to the idea of using the number five as a standard "break point."
So this is the ideal moment for you to start practicing multiplication with the higher digits in this additive fashion. Here I give just one example to illustrate the main idea:
7*8 = 7 * (5 + 3) = (7*5) + (7*3) = 35 + 21 = 56
Do not expect them to know what to do. Guide them with questions. You are supposed to pause and ask as you write:
“Eight is five plus what? Three? Is that correct? O.K.”
“Now, how much is seven times five? Yes, thirty-five, perfect! Thank you.”
“And, how much is seven times three? Yes, twenty one, very good!”
“So, now we just add those two numbers. Can you please add 35 + 21 ? Thanks.”
Just by listening to you asking them these questions and watching you as you write down this multiplication process step by step, they get it, they understand it, and they end up empowered by knowing they can get the right result by themselves even if they do not have memorized the result, even if it takes them a little while doing it in steps like above. They are now much closer to self-sufficiency when multiplying higher digits.

Multiplication Tables

A very important foundation for understanding math

Sometimes I see students who are really intelligent but who are having problems at school understanding new material. Some of these students only need to go over a few specific examples to grasp the concepts and move on to the next topic. So it is sad and almost unbelievable to discover that, in a few cases, the real obstacle standing on their way is that they do not know the multiplication tables! I remember when I was in second grade I hated learning the multiplication tables because the repetition process was so boring and it seemed meaningless to me at the time. However, in third grade I discovered the benefits of knowing by heart the multiplication tables. It allowed me to understand division. Understanding division allowed me to build a solid understanding of fractions.
In this time and age, many generations have grown up and gone through school using pocket calculators. A few people have made it all the way to college without ever learning how to multiply two numbers without using a calculator. The problem for them is, the more advanced the math courses they take, the more trouble they have at trying to figure out how formulas work by looking at specific examples. They cannot think their way through the examples because they don't know their multiplication tables; therefore their mastery of division, fractions and exponents is very limited and shaky.
To students who are preparing for the GMAT, GRE or CBEST, I always recommend to review, polish, extend and reinforce their knowledge of multiplication tables. The importance of this foundation cannot be stressed enough.