Mastering elementary mathematics can be compared with constructing and using a sturdy fishing net. Traditional elementary mathematics gives students lots of individual pieces of string - here's one string (addition) and another (subtraction) and one more (place value) and yet another (fractions), etc. But too often, students who master individual strings still do not possess a fishing net.
What will help unify the study of elementary math? Both parents and children need broad, practical, connecting principles that unify facts, skills, procedures, and concepts. We need a frame upon which to attach all the pieces of string, and we need strong knots that remind us how various strings are connected. The principles in this series act as the knots and the frame that tie math together, providing connections that endure across grades and topics.
Name It and Claim It
Last issue's article focused on the principle that "Quantities have first and last names." Quantities like "400" or "4/7" are read as "four hundred" and "four sevenths," respectively. Each quantity has a first name and a last name. The first name of "400" is the first thing that is said - "four" - and the last name is the last thing that is said - "hundred." The first name is how many; the last name is what it is. Learning that quantities have first and last names sets students up for success in almost every topic of math: place value, fractions, decimals, geometric measurement, and even algebra.
Skip, Skip, Skip is the Glue
A second connecting principle for elementary mathematics is skip counting - counting by multiples of a number, such as "2, 4, 6, 8, 10, 12, . . ." or "7, 14, 21, 28, 35, . . ." Skip-count mastery can be greatly helped by the use of "skip-count songs" - tunes and lyrics which teach children how to count in multiples, from 2 up through and including 10. Skip counting reveals and clarifies much of the basic structure and order inherently present in the field of whole numbers, which in turn leads to a more solid and connected mastery of the basic arithmetic operations (+, -, x, /).
Skip-count songs draw upon children's imaginations in memorizing basic arithmetic sequences. Such songs can be played and sung during the opening minutes of a math lesson, at transition times during the day, or at home or in the car. Just as children repeat throughout their day the "ABC" song or catchy tunes from commercials, so also they can "practice" the skip-count choruses while swinging, playing catch, riding to the store, etc.
Skip-count songs are not addition or multiplication facts per se. Rather, the basic facts are embedded within the ordered lists in the songs' choruses, which teach how to "group count."
The skip-counting process provides a rich mathematical environment for basic arithmetic operations. Skip-count songs can be connected with several other teaming modes, such as manipulatives and real-world objects. For example, I teach children to use their fingers as they skip count. Students raise one finger after another in the usual counting fashion, except that they are now counting by multiples: "3 [first finger up], 6 [second finger], 9 [third finger], 12 [fourth finger], 15 [fifth finger] wheels go round and round . . ." This skip-counting process connects with both addition and multiplication. The songs show addition (e.g., 3 plus 3 is 6, then 6 plus 3 is 9, then 9 plus 3 is 12, etc.) as well as multiplication (e.g., counting 3 four times was 3, 6, 9, 12 - so 3 four times is 12).
The skip-counting connection with basic addition facts is a skeleton upon which addition facts can be hung. From the 3's chorus alone, a child can connect the following basic single-digit addition facts: 3 + 3 = 6; 6 + 3 = 9; and 9 + 3 = 12. But the benefits of skip-count choruses go beyond mastery of single digit basic addition. Embedded in the 3's chorus are further addition facts: 12 + 3 = 15; 15 + 3 = 18; 18 + 3 = 21; 21 + 3 = 24; 24 + 3 = 27. For a child who has memorized the skip-count choruses, an addition exercise such as 19 + 3 can be connected to the 3's skip-count chorus sequence which identifies that 18 + 3 = 21, so 19 + 3 is just 1 more, or 22. These types of connections are even more plentiful in the "higher" choruses, such as the 9's: 9 + 9 = 18; 18 + 9 = 27; 27 + 9 = 36; 36 + 9 = 45; 45 + 9 = 54; etc.
One other powerful connection for skip counting is the rectangle/area model - another organizing principle which will be examined in a separate article. The rectangle/area model is an extremely versatile model that uses unit squares grouped in rectangular form, where the length, width, and area of the rectangle can be considered. Because of their horizontal rows and vertical columns, rectangles virtually beg to be skip counted. It is easy to count how many items are in each row, and then simply skip count the rows.
Figure 1: Whisper counting with the Rectangle Area Model
This can be done by introducing "whisper counting," as shown in Figure 1. I begin by having students count across by 1's the number of squares in the rectangle, each time through saying the faint numerals more and more softly, and the bold numbers more and more loudly, as we point at or touch each number. Eventually, the only numbers being said aloud are the boldface numerals on the far right of the diagram, which is the chorus for the 4's song. Similar practice can be done with each of the other numbers from 2 up through 10.
Although multiplication is closely related to children's ability to think additively, research shows that multiplication requires higher-order thinking that essentially involves groupings. Mastery of skip-count songs establishes a firm foundation for such higher-order thinking. With a variety of activities such as skip-count songs, finger counting, and whisper counting, children learn that skip counting is a short cut for counting by ones. This builds up students' confidence in counting by groupings, which is fundamental for multiplication.
Another connection I use is color-coded tagboard strips containing the numbers of each chorus for students to reference. The numerals on the strips are color-coded to match the colors of the math manipulatives. I prefer Mortensen Math, so the orange numerals on my strip for the 2's chorus match the orange 2-bar, the pink numerals on the strip for the 3's chorus match the pink 3-bar, the yellow numerals on the strip for the 4's chorus match the yellow 4-bar, and so forth. Like team uniforms, this type of color coding helps children mentally associate appropriate numbers into the skip count groups to which they belong. Of course, some numbers like 6 belong to more than one "team," since 6 is on the chorus for the 2's, the 3's, and the 6's.
Writing and Reciting
Students can also engage in two other practices that reinforce skip counting fluency: writing the various choruses, and orally reciting them. Writing the numerals "3, 6, 9, 12, 15, . . ." helps solidify number sense kinesthetically via the movement of printing. Oral recitation - saying the choruses, not just singing them - is also important, since singing and speaking are two different processes, controlled by different parts of the brain.
Time and Money
The skill of skip counting links directly to two important successes I have observed in students. The first link is telling time from a traditional analog clock. Since clocks have the minutes marked in groups of five, students who know the 5's chorus can more easily skip count the minutes shown by the minute hand. If the long (minute) hand is pointing at the 7, students just count by 5 seven times: "5, 10, 15, 20, 25, 30, 35 - it's 35 minutes after the hour."
The second link is in counting money. I tape a penny to a green 1-bar, a nickel to a light blue 5-bar, and a dime to the dark blue 10-bar. Children now see a visual, color-coded connection that the nickel is worth 5 pennies. After teaching a lesson on money to first grade students who had already learned the skip count choruses, a fellow teacher remarked:
They had it clear as a bell. They know a five-bar, and they can count by five. I used a transparency of coins [3 nickels and 2 pennies] on the overhead and they counted, "Five, ten, fifteen ... sixteen, seventeen.' In no other year have I ever been able to go into that kind of addition grouping on the first day of dealing with money and get an understanding like that.
Divide and Conquer
Skip counting also prepares young students for success with division in later grades. Several years ago, I was working with a 7-year old student who had been listening to the skip-count songs for about a year and had learned all the choruses. One day, he asked me an age-appropriate question for a second grader, "What's division? I've never heard of that." I replied, "Division is counting how many of one amount are in another amount, so '12 / 3' means '12 has how many 3's?'"
He immediately said, "Four." I was astonished at his quick reply and asked him how he figured that out so fast. He shrugged and said with a smile, "I know all the choruses. What's the big deal?"
Wanting to test him, I then gave him about 5 or 6 division exercises, such as "32 / 8" or "54 / 9." He answered every one correctly (and quickly!) and he still had that confident smile on his face. Here was a child who had automated all the skip-count choruses and who now understood the idea of division, and he didn't think division was difficult at all. His earlier skip-counting experience was a set up for this later success with division. This same student has also had similar success with common denominators for fractions.
How early can children start learning skip-count songs? My younger son was listening to them before he turned 2. He simply enjoyed listening the music, and he learned to repeat the 2's chorus even before learning one-to-one counting correspondence.
Did he truly understand counting in multiples? At that early age, of course not! However, as he grew older and began formal study of arithmetic, it was clear that his acquired familiarity with the skip count choruses made understanding arithmetic concepts and skills much easier.
Is it enough to simply play the skip-count songs often, while doing any basic math curriculum? Certainly, because mastery of the skip-count songs will strengthen a child's fluency with our number system, which all curricula teach. More strengthening will occur with math curricula that specifically address and practice skip counting in a variety of contexts. Such skip-counting connections will be addressed in future articles.
Musical skip counting is a firm foundation for multiple successes in elementary mathematics. Let's teach our kids to skip count early and often. Skip counting will bring success and provide connections between the many different areas of elementary mathematics - you can count on it!