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 worked-out 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                                               Dot Diagrams a.  Dots:    (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.

 2(5) = 10

1)  number sentence ---- 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.

 1(21)

h) 6(2) + 3(3) = ___________.

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