Common Mistakes in Teaching Elementary Math—And How to Avoid Them

Fuchang Liu

• Publisher: Routledge
• Year: 2017
• ISBN:
  9781315511757
• Edition: 1
• Topic: Bildung
• Subtopic: Bildung Allgemein

Learn the most effective ways to teach elementary math, no matter how much experience you have with the subject. In this book, Fuchang Liu takes you through many common mistakes in math instruction and explains the misunderstandings behind them. He points out practices that should be avoided, helping you to adjust your lessons so that all students can achieve success.

You'll discover how to…

- Increase your confidence with core math principles and reasoning

- Set your students on the path toward eventually developing more complex math skills

- Improve student achievement by approaching problems in logical yet creative ways

- Overcome common challenges faced by students and teachers

- Teach problem solving for different learning styles

Every chapter reconsiders well-established ways of teaching all areas of elementary math, from addition and subtraction to statistics and graphs. Helpful examples and tips are scattered throughout the book, offering revisions to the way these topics are often presented in the classroom. Also included are group study ideas for principals and instructional coaches so your school or district can work on the book together. With this practical guide, you'll be ready to help students truly develop their math understanding.

1
Counting

Counting Shouldn’t Start at 0

“Boys and girls, let’s have a little practice on counting. Now, count after me. Zero! One! Two! Three! Four!” This is what Jane, the fictitious elementary school teacher in this book, directed her children to do in class one day. On a different occasion, as she found her hundred chart didn’t have a 0, she added it there herself, right above the number 10.

While 0 is indeed immediately before 1 in terms of the relative positions for whole numbers, having children count starting at 0 is problematic. Before we embark on further discussion of this issue, let’s look at two common kinds of numbers: natural numbers and whole numbers.

Natural numbers are just 1, 2, 3, 4…. Why do we call them natural numbers? It’s because such numbers came into being in the most natural manner. Many things in our lives are considered natural, for example, “natural languages” that came into existence with early human beings, as opposed to “artificial languages” created for a specific purpose. Within natural languages, some phonemes are more natural than others. For example, babies start to pronounce more natural sounds, such as /mɑ:/ before less natural sounds such as those containing /r/ or /l/. It’s probably not a coincidence that /mɑ:/, meaning mother, is among the very first few words babies acquire and is found in many languages around the world. Similarly, we can imagine that our ancient ancestors, when the need for numbers started to arise, invented means to represent 1, 2, 3…. Henceforth they have become known as natural numbers.

The number 0, in contrast, came into the picture very late. Its concept as used in modern times originated in the 7th century, and this number was introduced into the decimal system as late as in the 13th century (the Roman numeration system, the numerals often used on the face of analog clocks and opening pages of books, still doesn’t have a means for expressing 0). This indicates that the representation of 0 is for its “nothingness” instead of counting.

If we add this 0 to the set of natural numbers, then we have whole numbers. That is, natural numbers are 1, 2, 3, 4… and whole numbers are 0, 1, 2, 3, 4…. Unfortunately, the distinction between natural numbers and whole numbers is not unanimously agreed on. Sometimes 0 is included in the set of natural numbers. To avoid this confusion, numbers 1, 2, 3, 4… have come to acquire a new name: counting numbers.

It may become clear now. Counting numbers, as the term indicates, are numbers used for counting, and they start at 1. If you ask a child who has just learned the first 10 or 20 numbers to count, regardless of the language background that child is from, you will most likely hear “one, two, three, four …”, not “zero, one, two, three, four …”.

You may ask, “Are there detrimental consequences for counting starting at zero?”

There are two major problems with counting from 0. First, 0, with its meaning of “nothingness,” is very difficult for a young child just learning how to count (believe it or not, even for older children and adults retrieving multiplication facts, the reaction times are longer when a fact contains a 0, such as 3 × 0, than when a fact doesn’t, such as 3 × 4). Most young children learn their first numbers through a one-to-one correspondence between a number and the quantity of actual objects that number represents, as the number 1 and one toy soldier, the number 2 and two toy soldiers, and so on. Imagine, however, how you would teach a young child to call out 0 with no toy soldiers in sight.

Second, to determine the number of objects by counting, such as determining how many apples there are on the table, many children would touch or point to the first apple and say “one,” then move on to the second apple and say “two,” and continue in this manner until all the apples are counted (many adults do this too). If we start at 0, we would have to touch nothing and say “zero,” but then we would have to start touching apples and calling out “one, two, three,” and so on. This can be very confusing because there would be a need to stress when to touch (with a one-to-one correspondence) and when not to touch (without a one-to-one correspondence). If a child accidentally touches an apple while saying “zero,” then the total number of apples would be off by 1.

As a matter of fact, you don’t even have to look far to see what sequence is actually used in counting. The next time you find yourself counting anything—be it a wad of dollar bills, a group of children, or a number of apples as you put them in a bag when you want to buy 10 at a supermarket—see whether you’re saying “one, two, three …” or “zero, one, two, three …”.

What Are Tally Marks for?

To help her children practice counting, Jane did a number of different activities. One of them was to survey her children’s favorite types of pizza. She first wrote the few most common types on the board: “Cheese,” “Mushroom,” and “Pepperoni.” Then she asked, “How many of you like cheese pizza best?” Some children raised their hands. Jane led the whole class in counting the number of hands, recorded it with tally marks, and moved on to the next type. After the survey was taken, she had such results on the board (see Figure 1.1).

Jane’s use of tally marks, however, was inappropriate for this situation. This is because tally marks aren’t nearly as efficient in recording quantities as numbers are. To explore this a little further, let’s first take a look at the most common use of tally marks.

Suppose you were doing the same survey, but in a different fashion. After you wrote all the three types of pizza on the board, you asked all your children to go to the board, one by one, and indicate his or her favorite type of pizza under the corresponding label. During this process, numbers would be a poor choice to use. If the first child who preferred pepperoni pizza wrote a 1 under that category, the next child who also liked pepperoni pizza best would have to erase the 1 and write a 2 over it. Similarly, all later children whose favorite type of pizza was pepperoni would erase the previous number and then write a new number over it. This is certainly not an efficient way of recording the survey results. Tally marks, in contrast, fit nicely here. Whoever liked pepperoni pizza best would simply need to put a tally mark under the “Pepperoni” category, and all later children who favored pepperoni would simply need to add another mark under that category. Similarly, those whose favorite pizza was mushroom would simply need to put a tally mark under the “Mushroom” category. No erasing and rewriting would be needed. This is the ideal situation for using tally marks, a situation that requires constantly updating a number.

Figure 1.1 Jane’s Survey of Her Children’s Favorite Types of Pizza

Tally marks are essentially a one-to-one correspondence between the number of marks and that of actual objects or people (that is, the use of the number of marks representing as many real objects or people). The fact that four marks are drawn in one direction while the fifth is drawn in another is for easy counting at the end of tallying. With every fifth mark traversing the previous four to form a group of five, we can count by 5s instead of by 1s. That should be faster and less prone to error.

But can tally marks be used in place of numbers?

The answer is no. As the name indicates, tally marks are just for tallying, and usually for a small number at that. Outside this realm, they are almost useless due to their one-to-one nature, as opposed to the symbolic nature of Arabic numerals such as 4752 (imagine having to represent 4752 using tally marks, not to mention using tally marks to compute). Even for tallying purposes, they are inefficient. Suppose you had a large class and 48 children liked pepperoni pizza best. You would first have to count the tallies to find out. That is why people almost always write down the corresponding total number after tallying a count.

In the aforementioned survey where Jane’s children indicated their preferences with a show of hands, the total number for each category was already available when she counted them. She simply needed to write down each number directly. There was no need to draw tally marks. Only when she had to constantly update the number did she need to use tally marks.

It May Not Be Fast—the Purpose of Skip-Counting

“Skip-counting is for counting up to a number faster than counting one by one,” Jane told her children one day while teaching the topic of skip-counting. She engaged them in an activity in which they would count out 100 cubes from a pile, first by 1s, then by 2s, and then by 5s. At the end, she showed them the time she had kept so they could see which method got them to 100 the fastest.

First of all, we need to distinguish two types of counting: counting real objects in order to find out their quantity, and counting without any real objects in sight, usually for practicing purposes. Skip-counting is often used in the latter case for understanding the properties of certain numbers in a sequence.

When counting real objects, people may do it by different quantities at a time, such as by 1s, by 2s, and by 5s. You might be tempted to think that as the size to count by increases, the faster it’ll be to finish the counting task. This may be true for only a few sizes. Let’s suppose we want to count out 100 objects by 1s, 2s, 5s, and 10s to see if some methods are faster than others. Obviously, counting out objects by 2s is faster than by 1s. Other than that, however, counting by larger numbers probably won’t give you an edge, as counting out objects by 5s won’t necessarily be faster than by 2s, and counting out objects by 10s won’t necessarily be faster than by 5s. The reason is this: Counting objects by 2s doesn’t exactly involve counting out two objects sequentially and separately but rather the two objects are counted out at the same time. That is, for two objects, we don’t actually count them—we just immediately realize that there’re two such objects (this is known as subitizing, the perception of a small quantity of 1, 2, or 3 in a very fast manner without actually counting). Therefore, compared with counting by 1s, the number of objects counted out by counting by 2s at each round is doubled (“Two, four, six…”). But for numbers 5 and up, the counting can’t be skipped. You can easily imagine that counting by 10s won’t necessarily be faster than counting by 2s, because for each group of 10, you still need to count them out by a smaller number, such as by 1s or by 2s.

Numbers larger than 10 are rarely used as a group to count real objects by. (Have you ever tried counting real objects by 15s or by 40s?) Still, sometimes it’s a good idea to count out objects by a larger number, such as by 5s or by 10s, and leave them in separate piles. The purpose isn’t so much for speed as for breaking the whole counting process into several smaller chunks so that if you ever mess up one chunk, you don’t have to go all the way back to the beginning and start over again. Instead, all you need to do is recount that particular chunk and go on with the rest of the objects.

The other type of counting—namely, counting without real objects involved—is often used with young children, usually for the purpose of teaching them some sequences of natural numbers and their properties. It’s here that skip-counting is used. The following Math in Action box lists the three most common skip-counting sequences. As for skip-counting by 10s, it’s not very different from counting by 1s in that we just need to attach a 0 to each number in the sequence of natural numbers: 10, 20, 30 ….

Math in Action: The Three Most Common Skip-Counting Sequences

1. Skip-counting by 2s starting from 2. This will produce a series of even numbers: 2, 4, 6, 8, ….
2. Skip-counting by 2s starting from 1. This will produce a series of odd numbers: 1, 3, 5, 7, ….
3. Skip-counting by 5s starting from 5. Such numbers are multiples of 5, and they end in either 5 or 0: 5, 10, 15, 20, ….

Speed in this second type of counting usually isn’t a factor to consider. If speed were important, then counting up to 100 could take only two numbers if you counted by 50s: 50, 100. This certainly wouldn’t serve any useful purpose at all. Just like counting real objects, numbers larger than 10 aren’t often used as a group in counting without using real objects, either. For example, we normally wouldn’t ask our children to skip-count by 37s, or by 289s.

Our Number System Is Base-10, Not Base-12

“Boys and girls, let’s have a little practice on counting before we learn our lesson today. Now count with me: One! Two! Three! Four! Five! Six! Seven! Eight! Nine! Ten! Eleven! Twelve! Great! Let’s start over again.” This is how Jane started her lesson one day having her children practice on the first 12 counting numbers.

Counting numbers are 1, 2, 3, 4… and they go on and on. For obvious reasons, we don’t throw a long sequence of these numbers at young learners all at once. Instead, we introduce such numbers in chunks. We teach them several numbers at one time, then move on to the next several at another time, and this cycle is repeated until they can count up to a fairly large number. Typically, by the time children can count up to 100 and a little bit beyond, most of them can see the pattern in this counting sequence, and teaching them how to count after that isn’t quite necessary.

But for the beginning numbers, some consideration needs to be given to how much a chunk should contain. The English number words have some influence on us in this respect, that is, the first 12 words are each completely unique (one, two, three … ten, eleven, twelve) and then there is a series of teen-words (thirteen, fourteen, fifteen … nineteen). Moreover, there is a special word in English for 12: dozen. Reflected in real life, some merchandise is grouped in dozens, such as cartons of eggs and boxes of donuts sold in supermarkets. Due to such influence, Jane used 1 through 12 as the first chunk to teach. However, this isn’t ideal. To explore this issue, let’s look at the numeration system of the counting numbers we have today.

The numeration system used by most peoples over the world today is the Hindu-Arabic system. This system is what we commonly call the base-10 system, or the decimal system (deci means “ten”). In this system, 10 ones form a ten, 10 tens form a hundred, 10 hundreds form a thousand, and so on. In other words, in numbers written out in this system, each place has a value 10 times that of the place to its right. For example, the number 528 means 5 × 100 + 2 × 10 + 8 × 1. Put in another way, the place held by digit 5 is 10 times that held by digit 2, which in turn is 10 times that held by digit 8.

However, there is a huge obstacle to English-speaking children grasping this concept: the incongruity between Arabic number marks and the corresponding words in English. This is especially true for 2-digit numbers. This issue is reflected in several outstanding aspects. First, the number 11 means 1 ten and 1 one, but the corresponding English word eleven doesn’t have that indication (at least to people wit...