Christopher Horton, Ph.D.
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”.
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.
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.
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.
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.
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.
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.
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
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.
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
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.
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.
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
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.
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
– Activity 1 Name:
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)
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.
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.
Change the following word sentences into number
sentences and dot diagrams.
two groups of five dots is equivalent to ten dots.
1) number sentence ----
one group of two dots and two groups of one dot is equivalent to _______________
number sentence ----
Change the following number sentences into word
sentences and dot diagrams.
h) 6(2) + 3(3) = ___________.
word sentence ----
Six groups of two plus three
groups of three is the same as one group of twenty one.
+ 4(3) = _________.
word sentence ----