An Introduction to the Mathematics of Quantity
Christopher Horton, Ph.D.
Worcester MA (May 2008); chr1sh0rt0n@mac.com
Teachers who are working in middle and high schools with the program “Cognitive
Instruction in Mathematical Modeling” (CIMM), developed by Dr. Rob MacDuff and
associates, are enthusiastic about it and are reporting dramatic success. This program is
based on the mathematical system developed by Rob, referred to variously as Math
Modeling, “The Dots” and the “Mathematics of Quantity” or the “Mathematics of
Quantity and Relationships”.
The system starts with the idea of quantity as a representation of a collection of like
objects, and the concept of group as a structure imposed on a quantity. At the core of this
system is the concept of “number”, which has shifted from being an abstract entity
defined in terms of a collection of axioms and definitions to being a representation of
relationships between quantities or groups of quantities. Number so defined is thus
traceable back to a model of an aspect of physical reality, quantity, and derives its
character from its observable grouping properties.
The system that emerges from this new foundation is at once simpler and more powerful
than conventional mathematics. Its attributes and assumptions are further described and
explained somewhat in Dr. MacDuff’s essay “The Math Problem”, but if I were to
attempt a deeper and more complete explanation here, its meaning would most probably
elude the reader. Experience shows that this new system is much easier than the old for a
ten-year-old to learn and master, but for those of us who have spent years or decades
mastering contemporary mathematics it can present a challenge.
To say that this is a new system of mathematics doesn’t fully communicate how different
it is. A better explanation is that it represents a paradigm shift, a complete reordering of
ways of seeing. Well-known examples of such transformative changes in thinking
include Copernicus’ heliocentric theory, Einstein’s theory of relativity, Darwin’s theory
of natural selection and – for some readers - the Modeling Method of Physics Instruction.
It will be for future generations to judge whether the “mathematics of quantity” matches
these in importance, but it certainly resembles them in that it involves a profound change
in context and meaning, a change that has wide ramifications in the way many things in
mathematics and beyond are seen and interpreted. Such a change can only be understood
from within its own context and cannot properly be evaluated in terms of previous
systems of thought. The pilgrim seeking to understand must take the journey into this
new paradigm, preferably with the help of a teacher or fellow seekers, and must learn to
think and operate in it. Once properly understood, it becomes the way in which the world
occurs, and this cannot be undone.
Sharing my own experience may help explain my meaning. For twenty-five years I have
been teaching physics and mathematics in colleges and high schools, sometimes part-time
and sometimes full time. Always I have sought to understand what my students were
learning and how they were thinking, what I wanted them to learn, and why. During
three years of teaching remedial math to ninth graders in an inner city high school I
studied and experimented with a number of innovative approaches, and I developed and
tested my own instructional materials using arrays and groups of dots to represent
numbers. My journey led me to the Modeling Instruction Program at Arizona State
University, where I took Modeling Workshops in high school and college physics and
chemistry teaching and, in 2003, the course “Integrated Mathematics and Physics” led by
Dr. MacDuff, Dr. Richard Hewko and Colleen Megowan. Thus when I journeyed to
Arizona in the summer of 2007 to spend some time working with Rob and his colleagues
and to sit in on an introductory CIMM class for middle school teachers, I considered
myself well prepared to understand what he was saying and doing.
My progress, however, has not been simple or linear. Old ways of looking at things and
old habits in what I pay attention to have proven very persistent. Time and again I
believed and claimed that I understood something, only to discover that I had it wrong, or
even completely backwards. Having people to give me feedback and to challenge me to
look at what I am saying and thinking has been essential to my forward movement. The
years I spent mastering and using conventional mathematics have made this process
harder, not easier for me. For example, a lifetime of using number as an abstraction, a
simple mental object that can be illustrated by collections of physical objects, has made
learning to draw a distinction between number and quantity – and remembering to do so very difficult. It takes only a few minutes to explain the new conception of number as a
class of relationships between quantities. It is a simple and elegant conception, yet it
took me many months of working with number in the new system before I finally
understood that I didn’t really understand, and more months of struggle before I began to
see it clearly.
The difficulty of briefly communicating the thinking embodied in the “mathematics of
quantity” being what it is, I won’t attempt to present a survey of the entire system.
Instead I have reproduced the teacher notes from the first section of the first unit of the
first course for middle and high school students – Cognition Ignition, one course in
Cognitive Instruction in Mathematical Modeling or CIMM - together with a few workedout problems and a few that are left for the reader to work through. This will give you a
look at some of the foundational concepts and tools on which this system is built. It will
give you a chance to dip your toe in, to see whether you like it, whether you see a
possibility in it, and whether you might want to pursue it further.
Introductory Lesson on Quantity, Teacher Notes 2.1
R.MacDuff and R.A. Hewko, Copyright Info Dynamics Applications, 2007
The main purpose of this unit is to develop an understanding of number and its role in the
description of quantities. A quantity is, roughly speaking, something that has – or comes
in – units, such as three horses or four and one-half inches or five dots. Students think of
numbers as a stand-in for quantity, an abstraction of it. They will use this concept of
number as a starting place for understanding quantity, and, as they move through this
unit, finally arrive at a deeper understanding of number.
Each of the activities to follow involves dots, which can be used to represent any discrete
quantity. Although the students will be using numbers, surprisingly they will not be
seeing the numbers as a part of arithmetic. They will just see each numeral as a symbol
for representing different quantities of dots or different quantities consisting of groups of
dots. In fact when most students write down the numerical symbols these are not
symbols to represent mathematical concepts, but rather symbols to represent discrete
things. The students have not separated their concept of number from the concept of
quantities. It is important that the students discover the link between their descriptions of
objects, their own actions on objects and their concept of number.
This diagram can be thought of as being composed of both dots
and groups of dots. The object can be described as having two
groups, where each group has three dots. However the same
object can be regrouped into three groups, each having
two dots. For the student to understand the process by
which they can change the grouping structure and know
and understand what stays the same and what changes,
they need conceptual tools. These tools allow students to construct ideas that are relevant
to the problem. The ability to represent their ideas requires representational tools. These
representational tools provide ways of translating their ideas into a form that can be
considered and communicated.
In this section the students will learn to distinguish between dots and groups of dots (and
groups of groups of dots). They will learn to pay attention to the way the dots are
grouped, and will learn to represent these entities using numerals (the mathematical
symbols for number: 1, 2, 3, 4, … etc.), the operation + and the equality (=) relation.
Conceptual Tools
The basic ideas are that objects can be grouped into groups or multiple groups and that
the rearrangement of the groups or the number of objects within a group does not alter the
total number of objects.
The importance of dots is that every quantity can be represented by dots, a group of dots
or as multiple groups of dots. The students, by rearranging the same quantity of dots in
multiple ways, will become aware that the group structure does not influence the total
quantity of dots.
A description of the dots requires the student to pay attention to both dots and groups of
dots. There are three things that the student must be aware of when examining different
groups of dots:
1/ When are things “the same”? Things are the same when they have the same number
of groups and each group contains the same number of dots.
2/ When are things “not the same”? Things are not the same when the total number of
dots is not the same.
3/ When are things “the same but not the same”? Things are the same but not the same
when the quantity of dots is the same but the number of groups is different.
Representational Tools
Students will learn to represent each dot diagram, where circles indicate groups of dots,
in two different ways:
a) in words; -- two groups of four dots;
b) with a number sentence.
In a number sentence, a group of dots is represented by brackets ( ), and the number of
dots within the group is indicated by a number placed within the group symbol, e.g. (4).
A number placed in front of the group symbol, e.g. 2(4), indicates the number of groups.
For example, the diagram to the right can be described
in the following way: 2(4) = 8, which can be read as
“2 groups, where each group is made up of 4 dots”.
Many students will want to read 2(4) as “two times
four”. This should be discouraged initially, so that the student will focus on the meaning
associated with the mathematical phrase. (The language “two times four” will be used
later to describe the process of constructing the symbol for an equivalent group.) The
students will be surprised when they eventually discover that the descriptions they have
created can be interpreted in terms of multiplication!
Students will be led to articulate the insight that the order followed in describing the
grouping possibilities of a quantity – e.g. “three groups of two dots” or “two groups of
three dots” – is important. However while these both represent the same quantity of dots
they are nevertheless not the same thing, this is the concept of equivalence: the same
but not the same.
The initial activities are designed to tap into the students’ creativity. Thus students are
asked to be creative in writing as many different ways as possible of organizing a given
set of dots and representing those organizations using mathematical symbols.
It is important in the beginning to keep the different representations separated, as each
representation has its own syntax and rules. Mixing the representations can cause much
confusion, as it makes it extremely difficult for the student to learn the different syntaxes
associated with each language.
Consider the following example where there are eight dots. Starting out with eight dots
in total the student is asked to regroup the dots in a variety of different ways, and is asked
to represent what s/he has constructed using both symbols and words.
Representations
a. Dots:
Dot Diagrams
(Eight dots, no groups)
b. Word sentence: “Two groups of four dots
is the equivalent to eight dots.”
Number sentence:
2(4) = 8
c. Word sentence: “Four groups of two dots is
equivalent to eight dots.”
Number sentence:
4(2) = 8
d. Word sentence: “One groups of eight dots is
equivalent to eight dots.”
Number sentence:
1(8) = 8
(Students could also write 4+ 4 = 8, 2 + 2 + 2 +
2 = 8, etc.)
Students should be led to articulate the insight that the order followed – e.g. three groups
of two dots or two groups of three dots – is important. While they both represent the
same number of dots and are therefore “equivalent”, they have different grouping
structures, and are therefore “the same but not the same”. Some students will have a
struggle with this. Direct their attention to the grouping structure as being important.
Invite them to think of examples of things that are grouped differently but are equivalent.
For example, a 12-pack, two six-packs and twelve singles of 12-ounce cans of Pepsi are
equivalent – the same amount of Pepsi - but they won’t be mixed together on the shelves,
they won’t be “rung up” as the same thing at the cash register and they may not cost the
same amount.
Students will persist in using the word “equals” to describe the relationship between two
expressions that contain the same number of dots. Gently but persistently steer them
back to using the word “equivalent” since the focus to get students to see they are “the
same but not the same.”
Students will tend to skip writing the “word sentences”. These are an absolutely essential
part of the description process and of the students’ development of their ability to think
about mathematical concepts using multiple representations. Keep reminding them not
to skip the written part of an answer.
Simple Grouping – Activity 1 Name:
Objectives: After completing this section the student will understand:
1/ a group can be regrouped in multiple ways. 3(2) = 2(3) = 1(6) = 6(1)
2/ regrouping does not affect the total number. 3(2) =6= 2(3)
Carefully consider each set of dots given below. Place a circle around dots to form
groups in as many different ways as possible. Describe each regrouping using both
mathematical symbols and words. Pay close attention to the groups and the dots
within each group.
a)
Various answers: Look for patterns in the answer, and help students to see new
arrangements such as (a) all groups having an equal number of dots, (b) 10 groups of one
dot, (c) 1 group of ten dots, (d) one group of seven dots with 3 dots left over, (e) five
groups of zero dots and one group of 10 dots, (f) ½ group of 20 dots, (g) all groups
different, etc. Try to help students to be creative and remove self-imposed blockages to
learning.
c)
Change the following word sentences into number sentences and dot diagrams.
d) two groups of five dots is equivalent to ten dots.
1) number sentence ---2(5) = 10
2) dot diagram
=
f) one group of two dots and two groups of one dot is equivalent to _______________
1) number sentence ---2) dot diagram
Change the following number sentences into word sentences and dot diagrams.
h) 6(2) + 3(3) = ___________. 1(21)
1) word sentence ---Six groups of two plus three groups of three is the same as one group of twenty one.
2) dot diagram
i) 1(5) + 4(3) = _________.
1) word sentence ----
2) dot diagram