Teacher-Directed Counting
October 22, 2019
A deep understanding of the number line is essential to understanding and succeeding in elementary mathematics. Skip counting while changing directions is an efficient method to gain a mastery of the number line, because it demands that students comfortably shift directions, incorporating adding, subtracting, multiplying, and dividing into one focused exercise.
Although choral counting is a staple of most elementary school math routines, it’s rarely facilitated as a mathematical exercise. Instead, it normally consists of students reciting numbers in ascending and/or descending order, e.g.
0 1 2 3 4 5 6 7 8 9 10
10 9 8 7 6 5 4 3 2 1 0
or
0 3 6 9 12 15 18 21 24 27 30
30 27 24 21 18 15 12 9 6 3 0
Mastering this linear counting form is no more of a math exercise than reciting the U.S. Presidents in order is a history activity. Students can easily internalize a number set’s start and endpoints, as well as the order in between. However, when asked to make a transition within the sequence, e.g. What’s 1 more than 5?, the student who has only memorized numbers to ten as a linear chant likely needs to start back at one to find the answer, similarly to the way they would need to access the second to last line of the Pledge of Allegiance by beginning with the oath’s first sentence.
Linear counting with a defined start and endpoint assigns the same importance to each interval, because every number in the sequence is acknowledged equally. All skip counting transitions, however, are not equal in difficulty and thus shouldn’t be practiced the same amount of time. The child who needs excessive think time to answer What’s 1 more than 5? might not have the same struggle when asked What’s 1 more than 1? Therefore, they should practice the five-six transition more than one-two.
The same principles apply to other units, e.g. threes.
0 3 6 9 12 15 18 21 24 27 30
The 0 to 3, 3 to 6, and 6 to 9 intervals tend to be relatively simple for second and third grade students, as are the jumps from 12 to 15, 15 to 18, 21 to 24, and 24 to 27. The 9 to 12, 18 to 21, and 27 to 30 transitions, however, are much harder, because the tens value changes. Therefore, to master skip counting by threes the difficult transitions should be emphasized.
The vast majority of children don’t possess the wherewithal and/or discipline to recognize – and try to improve - these weaknesses. Thus, it is up to the teacher to address student weaknesses through deliberate practice.
When teaching a beginner to play the guitar, the instructor will direct their student to spend more time practicing a difficult chord transition, e.g. D to C, than they will G to e minor, because mastering the dexterity and rhythm of the former requires far more repetition than achieving it with the latter. Math teachers can use a similar approach through choral counting, cuing their students to count higher when they point up, to stop when they hold their hand sideways, and to count lower when they point down.
A concrete teacher-directed approach can be used for students who haven’t yet memorized the number sequence. To accomplish this, instructors construct a number line left to right across the students’ vision using their fingers.
Unlike the traditional American method of holding up an index finger, the teacher holds up their right pinky to represent one, right ring finger for two, right middle finger for three, and so on until their left pinky represents ten (see images on right).
Instead of asking students to count, teachers direct their class to say how many fingers they see as they push their hand forward.
This counting drill can be simplified further if teachers use a “counting glove” as a piece of theater, writing numbers on the corresponding fingers, so that the activity becomes an exercise in number recognition (see image on right).
Initially, children often struggle with teacher-directed counting, because it requires them to focus in two directions on the number line, and they’re not used to sustaining the intense concentration needed to succeed in the activity. On the first day of school, a teacher can usually hold their classes’ collective focus for about 30 seconds before students become exhausted by the exercise. With routine practice, however, children’s stamina improves and holding focus for several minutes becomes the norm. After a class is collectively fluent, the chorus creates a contagious rhythm and a friendly counting joust between teachers and students emerge. The former tries to stump the latter with increasing complexity while the latter aims to stay on-balance throughout the dance.
Best of all, once their endurance is built, students’ latent focus abilities permeate not only Math lessons, but their other studies as well.
Any meaningful teacher-directed counting exercise falls into one of two categories:
Constructing a number line.
Counting with Equivalency on the number line.
Constructing a Number Line
Constructing a number line is the process of building fluency within a set of units using previously learned strategies. The teacher’s role is to provide students with intensive practice so that the transitions become simple. Drills start slowly and gradually lead to fluidity, using a Mastery before moving on approach. Kindergarten teachers make sure that 90% of their students can count forward and backward, shifting directions to three before moving on to four. Then, they repeat the process, building fluency to four before moving on to five and so on until they’ve reached ten. Second grade teachers lead their students to master counting twos to eight before building their fluency to ten, e.g.
2 4 6 4 6 8 6 8 6 4 6 8 6 8 6 4 2 4 6 4 2 0
Foundations need to be strong for a tower to rise, and low multiples must be mastered before moving on to higher ones. Once students have constructed a mental number line for a set of units, they can apply this strategy to build fluency that complements the math topics they’re learning.
Counting with Equivalency on the Number Line
Once children can count and effortlessly change directions within a given number set, they are fluent with that portion of the number line. Teachers can channel this mastery by incorporating equivalency into number line counting, using simple to complex linear progressions. In doing so, students bolster their mathematical vocabulary, while more clearly seeing interconnectedness within the discipline.
The following dialogue could serve as a first grade fluency drill following a lesson in which students concretely learned units of ten within 100.
T: Count to 10 starting at zero.
S: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
T: Count to 10 tens starting at zero tens.
S: 0 tens, 1 ten, 2 tens, 3 tens, 4 tens, 5 tens, 6 tens, 7 tens, 8 tens, 9 tens, 10 tens.
T: Count to 10 tens again. This time, when I raise my hand stop. Go.
S: 0 tens, 1 ten, 2 tens, 3 tens.
T: (Raise hand.) Say 3 tens in standard form.
S: 30.
T: Continue.
S: 4 tens, 5 tens, 6 tens, 7 tens.
T: (Raise hand.) 7 tens in standard form is?
S: 70.
T: Continue.
S: 8 tens, 9 tens, 10 tens.
T: (Raise hand.) Standard form?
S: 100.
T: Count backwards in standard form, starting at 100.
S: 100, 90, 80, 70, 60.
T: (Raise hand.) How many tens are in 60?
S: 6 tens.
T: Continue.
S: 50, 40, 30, 20.
T: (Raise hand.) How many tens?
S: 2 tens.
T: Continue.
S: 10, 0.
Knowing when to stop the progressions and signal for an equivalent form is a rhythm and balance that teachers develop. Educators who feel some or all of their students are ready for additional complexity, might direct them to count by alternate forms, e.g.
1 ten 20 3 tens 40 5 tens 60 7 tens 80 9 tens 100
10 tens 90 8 tens 70 6 tens 50 4 tens 30 2 tens 10 0
A middle grades equivalent counting exercise might follow the progression below:
The complexity grows with each passing line. Like a ladder, a child needs to grasp each rung before reaching the top. Starting too complex leaves students frustrated and confused. However, building a solid foundation in which they master each level before moving on to a harder one helps children reach greater complexities than they or their teacher might imagine.
The process of ascending these five steps might take several minutes or days. It all depends on the students’ fluency with the topic.
Teachers begin at a level that all students feel confident answering correctly and pull (without pushing) the class to the highest level that they will collectively feel challenged, but not defeated. This exercise might be delivered as follows:
T: Count to 8 starting at zero. (Write as students count.)
S: 0, 1, 2, 3, 4, 5, 6, 7, 8.
T: Count to 8-fourths starting at zero-fourths. (Write as students count.)
S: 0-fourths, 1-fourth, 2-fourths, 3-fourths, 4-fourths, 5-fourths, 6-fourths, 7-fourths, 8-fourths.
T: Which fraction is equal to 1 whole?
S: 4-fourths.
T: (Cross out and write 1 beneath it.) Which fraction is equal to 2 wholes?
S: 8-fourths.
T: Cross out and write 2 beneath it.
T: (Move to the side of the board so that the number line is not in the center of the students’ vision.) Let’s count to 8-fourths again. This time, say the whole numbers. Try not to look at the board.
S: 0-fourths, 1-fourth, 2-fourths, 3-fourths, 1, 5-fourths, 6-fourths, 7-fourths, 2.
T: (Point at .) Say 5-fourths as a mixed number.
S: 1 and 1-fourth.
T: (Beneath 5/4 , write 1 1/4 .)
Repeat process for 6/4 and 7/4.
T: (Move to the side of the board so that the number line is not in the center of the students’ vision.) Let’s count to 8-fourths again. This time, say the whole numbers and mixed numbers. Try not to look at the board.
S: 0-fourths, 1-fourth, 2-fourths, 3-fourths, 1, 1 and 1-fourth, 1 and 2-fourths, 1 and 3-fourths, 2.
T: (Point at 2/4.) Say 2-fourths simplified.
S: 1 half.
Repeat process for 1 2/4.
T: (Move to the side of the board so that the number line is not in the center of the students’ vision.) Let’s count to 8-fourths again. This time, say the simplified fractions. Try not to look at the board.
S: 0-fourths, 1-fourth, 1-half, 3-fourths, 1, 1 and 1-fourth, 1 and 1-half, 1 and 3-fourths, 2.