How Students Learn:
Whole-Number Sense in the Primary Grades1
Foreword
This is the third of five RSS items regarding the 2005 NRC publication, How Students Learn: Mathematics in
the Classroom. RSS #43.a introduced three general learning principles that are the framework of How
Students Learn.
 Learning Principle 1: Teachers must engage students’ misconceptions.
 Learning Principle 2: Understanding requires factual knowledge and conceptual frameworks.
 Learning Principle 3: A metacognitive approach enables student self-monitoring.
In this summary, the three learning principles are treated in the order 2, 1, 3 and are applied to the teaching of
whole-number sense in the primary grades (P-2, students aged 4-8). The reordering of the learning principles
is to match three comparable teaching principles developed by the author. 2, 3
 Teaching Principle A: Know where you want to go (in terms of the knowledge that all students should
acquire in grade X).
 Teaching Principle B: Know where you are now (in terms of the knowledge students bring to your
classroom and that you can build upon).
 Teaching Principle C: Know the best way to get where you want to go (in terms of learning
opportunities, including metacognition, that you will provide to all students).
Fostering the Development of Whole-Number Sense: Grades P-2 (Students Aged 4-8)
Teaching Principle A: Know where you want to go
Deciding what knowledge to teach calls for explicit action in identifying what the National Council of Teachers
of Mathematics (NCTM) has referred to as “worthy mathematical tasks.” 4 Of all the mathematics that one
might try to teach, what mathematics is (a) appropriate at a given grade/age level and (b) prepares students
for the “worthy mathematical tasks” to come? Beginning with the observation that NCTM identifies number
sense as a major learning objective and that number sense has a “privileged position” on school report cards,
the author conducted studies that confirm that number sense is a central structure in understanding the whole
number system. By “central structure,” the author means “(1) that the structure is vital to successful
performance of a range of tasks … and (2) that future learning in these tasks is dependent on the structure.”
The Number Knowledge Test used in the study and study results are reported to establish that number sense
is, indeed, a central structure in the study of the whole number system.
Teaching Principle B: Know where you are now
The obvious purpose of knowing where you (i.e., the students) are with respect to number sense is to be able
to build on students current understandings. Grade levels are one way to cut the knowledge pie. However, it
is “equally important to be aware of the mistakes, misunderstandings, and partial understandings that are
typical for children at this age level.” To this end, the discussion of children’s understandings is organized by
age level. The Number Knowledge Test can be used at the beginning and end of a given school year, and the
results can be analyzed at each age level (as is done in the text for ages 4-8). As a result of having used the
test with hundreds of children, the author reports three insights that have influenced her teaching.
 Mathematics is about quantity, not number. Numbers are the tools used to describe, predict, and
explain quantities and quantity interactions (e.g., joining, separating, ordering). For this reason, it is
“crucial to introduce numbers to children in the context of quantities (objects) and quantity
representations (e.g., pictures of objects, dot set patterns, number lines, thermometers, dials) that
given the numbers meanings as quantities.
 The words of oral counting are the link between the world of quantity and the world of formal
symbols. Numbers are expressed orally as number words and in writing as formal symbols
(numerals). Counting in the context of exploring quantity builds up a network of meanings for each
number word. Then opportunities must be to connect the meanings to symbols representing numbers.
 Understanding number is a lengthy, step-by-step process. Understanding number is not an “aha”
experience. From the teacher’s point of view, it is step-by-step process – a series of lessons through
which student understanding moves up one level at time.
Teaching Principle C: Know the best way to get where you want to go
To illustrate how number sense can be taught, the text draws upon the 200-plus activities of the P-2 program
Number Worlds.5 Discussion is organized around six design principles of that program.
1. Exposing children to major forms of number representation. The five major forms are objects,
dot set patterns (as on dice or dominoes), segments on a line, segments on a scale (a thermometer or
bar graph), and segments or points on a dial. Games are used to explore each of these forms at each
grade level P-2.
2. Providing opportunities to link the world of quantity with the world of counting numbers and
the world of formal symbols. Two cartoon characters – Plus Pup and Minus Mouse – are used to
build conceptual bridges between the world of quantity and the world of counting numbers (by
developing the concepts of “adding one” and “taking one away”) and between the world of counting
numbers and the world of formal symbols (from “adding one” to +1; from “taking one away” to -1).
3. Providing visual and special analogs to number representations. The Skating Game is
associated with the dial representation of number. In the illustration, it is used to build the concepts of
counting up, counting down, and place value. Through a second game (Rosemary’s Magic Shoes), a
linear neighborhood of 100 side-by-side houses extends the three ideas to larger numbers.
4. Capturing students emotions and imaginations in the study of number. In the Dragon Quest
game, students become heroes by putting out the dragon’s fire or by rescuing fellow students that
have been captured by the dragon. To do this, the students must master the game’s use of numerals
and operation signs as they move along the game board and accumulate/lose and keep count of the
number of buckets of water required for the task.
5. Provide opportunities for students to acquire both computational fluency and conceptual
understanding. The Sky Land Blastoff game uses a specially designed thermometer in which
counting up and counting down are visually represented by the upward and downward movement of a
ribbon that represents the mercury in the thermometer. A Guess My Number game requires students
to strategize in the use of a single question to guess “my number” from among three whole numbers
bounded by two given whole numbers.
6. Encouraging the use of metacognitive processes that will facilitate knowledge construction.
This design principle moves through three levels of support. Level 1 support includes: question cards
developed for specific states of each small-group game (e.g., for the Dragon Quest game, “How many
buckets of water do you have now?”); follow-up questions (e.g., “How did you figure that out?”); and
transition of questioning from teacher to student through the assignment of the role of group Problem
Poser. Level 2 support includes more general questions, not specific to a game (e.g., “Who as gone
the farthest? How do you know?). Level 3 makes use of a Wrap-up Period in which a group’s
Reporter for a small-group problem-solving session is called upon to describe to the entire class both
the activity that the group did and what the group learned.
This section Teaching Principle C concludes with the report of the results of a experimental (control group)
study of the outcomes of the use of the six Number Worlds design principles.
About the Publisher
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Sciences, the National Academy of Engineering, the Institute of Medicine, and the National Research Council, all operating under a
charter granted by the Congress of the United States. The NAP publishes more than 200 books a year on a wide range of topics in
education, science, engineering, and health. The institutions represented by the NAP are unique in that they attract the nation's leading
experts in every field to serve on their award-winning panels and committees. http://www.nap.edu
Caveat Emptor
This report summary was prepared by Bob Kansky (robk@tribcsp.com). It’s one of a series offered to business, education, and policy
leaders who are interested in the systemic improvement of mathematics and science education. Readers are encouraged to consult the
original documents for further information.
National Research Council. (2005). How students learn: Mathematics in the classroom. M. S. Donovan & J. D.
Bransford (Eds.). Washington, DC: The National Academies Press. 272 pages.
http://newton.nap.edu/catalog/11101.html
2NOTE: The text/author does not use the term “Teaching Principle,” but its use here is consistent with what is written and
aids in making the distinction with the earlier learning principles.
3NOTE: This summary skims 52 pages of detailed examples of (and discourse on) the teaching principles, with references
to results of research conducted by the author. The summary is, at best, introductory. Also, although the focus is on
grades P-2, the teaching principles are applicable at other grades levels.
4National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics.
Reston, VA: Author.
5Griffin, S., & Case, R. (1996-2000). Number worlds. Durham, NH: Number Worlds Alliance. [NOTE: The materials
were published over the years 1996-2000 and are specific to each of the grades P-2.]