Noticing number patterns.
Repeating number series, Number order, Number patterning
The process of pattern recognition facilitates learning and significantly
enhances academic performance by enabling students to predict outcomes,
to organize the world around them, and to establish relationships
- Analyzing math problems to introduce operations
- Constructing a house
- Designing art work
- Determining numeric probability
- Observing nature
- Planning for retirement
- Planning to purchase a home
- Predicting a recession
- Predicting g.p.a.
- Projecting weight loss or gain
- Solving math puzzles
- Understanding the national debt
Students will be able to:
- Name and give examples of different types of recurring numeric
patterns (see "Background Information on Numeric Patterns").
- Recognize numeric patterns in their environment.
Students will be able to:
- Reflect upon their thinking processes when using this skill and
examine its effectiveness.
- Analyze the relationship among adjacent numbers. Look for a recurring
- Hypothesize a pattern structure.
- Test your hypothesis (see "Background Information on Numeric
- If a pattern does not appear, look for a different pattern (see
"Background Information on Numeric Patterns").
- Repeat steps 1-4 as necessary.
- Reflect upon the thinking process used when performing this skill
and examine its effectiveness:
- What worked?
- What did not work?
- How might you do it differently next time?
- Debrief - review and evaluate process, using both cognitive
and affective domains to achieve closure of the thinking activity.
- Metacognition - the act of consciously considering one's
own thought processes by planning, monitoring, and evaluating them
(thinking about your thinking).
- Pattern - an organizational arrangement
- Fibonacci series - see "Background Information on Numeric
Possible Procedure for Teaching the Skill
- Define the skill and discuss its importance.
- Introduce and model a repetitive numeric pattern from "Background
- Practice the pattern.
- Repeat steps 2 & 3 with each numeric pattern from "Background
Information on Numeric Patterns," providing activities that allow
discrimination among patterns learned.
- Debrief: Discuss techniques students used to arrive at conclusions.
Include both triumphs and tragedies (facilitations and roadblocks).
Note: A general strategy can be given to students that will help
them identify which type of series is used (these are explained,
in detail, in "Background Information on Numeric Patterns."
- If the series increases or decreases rapidly, look for a multiplication,
division or exponential series (the V technique).
- If the series does not increase or decrease rapidly, apply
the next level of the V technique, looking for a solution.
- If no solution can be found, look for a combined addition
and multiplication series.
- If no solution can be found, look for a Fibonacci series.
- If no solution can be found, look for another pattern.
- Define repetitive numeric patterning and discuss its purpose.
- Place ten piles of beans in a line; the first pile having one
bean, the second having two beans, the third having three beans...
and so on up to ten beans in the last pile.
- Ask students to count the beans in the first and second pile to
compare the numbers.
- Repeat #3 with the second and third pile, the third and fourth
- Explain to students that adding one bean each time creates a repetitive
- Tell students that being able to identify patterns can help them
- Go through skill steps with students and show them how skill steps
apply to figuring out the bean pattern.
- Repeat bean procedure with a +2 pattern. Help students apply the
skill steps to figure out the pattern.
- Debrief students on the process, the definition, and the importance
of this skill.
Integrating the Skill into the Curriculum
Understanding of non-linguistic recurring numeric patterns can be
facilitated by using the overhead projector and charts showing the
numbers 1 to 100 in boxes on the chart. Involve the class in finding
and describing numerical patterns on the chart. Begin by covering
the chart with a blank transparency and then marking all the even
numbers with a blue dot. Ask the class to explain how you arrived
at the resulting pattern. Depending on their experiences, they may
explain it as "plus two" or "even numbers." Remove the marked transparency.
On a clean transparency, mark in yellow the numbers in intervals of
three. After discussion, overlay the two sheets and note that +2 and
+3 addition patterns emerge. Mark these points with a red square,
and ask students if they can discover a new pattern. This can be continued
with the class, and then with clean, duplicated copies of the 1-to-100
chart allow students to construct their own patterns. Have them compare
and check identified patterns with fellow students.
Introduce and teach recurring numeric pattern recognition both as
an interesting intellectual skill and as a test-taking device. Teach
students the numeric problem-solving processes and the common types
of problems they are likely to encounter. Ask students to create problems
using two or three of these common types plus one or two types that
they make up. Have students exchange their "made-up" problems and
try to solve each other's.
Using a transparency or ditto of a 1-to-100 chart, blank out a pattern
of numbers. Ask students to fill in missing numbers.
Have a student describe the weather (weather chart, calendar).
Patterns can be found in all forms of non-linguistic information.
The more students can "see" patterns in non-linguistic information,
the more they can organize their world. This skill can assist the
brain with its natural tendency to organize information into patterns.
The ability to perceive repeating patterns in various settings is
important for the learner. Aptitude tests frequently include problems
involving numeric patterns. Everyday problems can often be solved
by recognizing the recurring patterns within them. Understanding that
the environment contains countless examples of obvious and subtle
repeating patterns introduces the learner to the elegant nature of
There are many types of non-linguistic patterns. In this unit we
will consider a few of those patterns--those commonly found on aptitude
tests and other forms of tests. However, if students are not presented
with anything more than this procedure, all they are learning is a
relatively sophisticated test-taking technique. The ultimate goal
of having students solve number series problems is to have them identify
different types of numeric patterns. Students should be encouraged
to find other types of patterns in numeric series. We will consider
numeric patterns, especially those identified by David Lewis and James
Greeene (Thinking Better, New York: Holt, Rinehart and Winston, 1982)
as important for aptitude tests.
More Background Information about Numeric
Jacobs, Harold R. Mathematics: A Human Endeavor. New York.: W.H.
Freeman and Co., 1982. Chapter 2, pp 57-118.
Lewis, David and James Green. Thinking Better. New York: Holt, Rinehart,
and Winston, 1982.
Marzano, Robert J. and Daisy E. Arredondo. Tactics for thinking.
Colorado: Mid-Continent Regional Educational Laboratory, 1986.