**Cognitive Instruction in Mathematical Modeling
(CIMM)**

(An Approach to Mathematics Education)

Robert MacDuff, Ph.D., President, InfoDynamics Applications Ltd.

(Phone) 480-205-6135

*America's high
schools are obsolete . . . By obsolete, I mean that our high schools—even
when they're working exactly as designed—cannot teach our kids what they
need to know today . . . This
isn't an accident or a flaw in the system; it is the system.* — Bill Gates, 2005

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The need for a new approach in mathematics education is unavoidable, as years of trying have confirmed. Trying harder, longer and louder is not getting the job done. CIMM is an acronym for a program of mathematics instruction that addresses this problem. It is based on a cognitive theory that connects reasoning processes with mathematical expressions, and it constitutes a paradigm shift in that it represents a different way of thinking about what it means to know and understand mathematics. As its name implies, it is an approach based upon cognitive science, and it includes a different way of understanding the nature of mathematics itself. This theory provides the underlying structure for a program that actively engages students in doing, thinking, talking, writing and reasoning with mathematical ideas. The impact on student attitudes and achievements is dramatic, even difficult to comprehend (see appendix A).

The CIMM program (what is different):

The CIMM program defines mathematical competence as the ability to:

1) **visualize**
mathematical expressions as representations of contextual situations,

2) **guide** the
manipulation of mathematical expressions by both axioms and context,

3) **express** mathematical
ideas in multiple languages,

4) **construct** and
interpret mathematical expressions.

The CIMM program places a strong emphasis on development of the following:

**Conceptual understanding: **

Rationale: Students prefer to guide their
mathematical operations by understanding conceptually what is happening and
using this understanding to select and apply the appropriate axioms or
algorithms. This understanding is
developed in a unique way with the use of *conceptual tools*, sophisticated tools to think with.

**Linguistic** **understanding**: the syntax and rules, or algorithms for constructing symbols.

Rationale: The ability to read, understand and construct mental models of what the symbols are referring to requires that students acquire a deep understanding of the linguistic aspects of mathematical languages and how these languages encapsulate the structural nature of what is being described.

**Conceptual tools:**
sophisticated tools for constructing mathematical concepts, introduced
throughout the curriculum.

Rationale: CIMM provides students with the conceptual tools necessary to build each new mathematical concept. It is through use of conceptual tools that students actually build their own mathematical understanding. These tools can be used to build an understanding of mathematics, but they can be used in other fields as well. These then become the support structure that enables transfer of concepts.

** **

**Representational tools**: multiple languages to express mathematical ideas.

Rationale: The different mathematical languages (languages that can be used to express mathematical concepts) each highlight different aspects of what is happening conceptually and thus enhance conceptual understanding. CIMM requires each student to symbolize their thinking in four formal languages - words, numbers, symbols and a diagrammatic language.

**Modeling pedagogy:**
Constructing and deploying models.

Rationale: An effective program requires a tight connection between the curriculum, content materials and pedagogy. CIMM utilizes the modeling pedagogy developed over twenty years in the Modeling Instruction Program at Arizona State University. The latter’s focus on models, Socratic dialogue and student presentations has been shown to greatly facilitate student thinking and reasoning skills. See http://modeling.asu.edu .

A comparison of standard and CIMM
approaches to development of mathematical understanding:

Cycles |
Standard |
% |
CIMM |
% |

Concept development (conceptual tools) |
no |
0 |
yes |
~30 |

Symbolization (representational tools) |
no |
0 |
yes |
~30 |

Manipulative skills |
yes |
~80 |
yes |
~20 |

Problem solving |
yes |
~20 |
yes |
~20 |

Many of these terms have a different meaning in CIMM, so they need some explication.

**Concept development. **Conceptual understanding generally implies knowledge
that a particular operation will produce a correct result based upon
application of a rule or axiom; e.g., “a minus times a minus is a
positive”. Conceptual
understanding in CIMM, by contrast, refers to the ability of the student to
interpret and visualize the referent of each mathematical symbol. Mathematical sentences become
propositions that refer to real-world situations. This eliminates the need to emphasize things such as the
order of operations, since the order in which the operations need to be
executed in any mathematical sentence is intuitively obvious from the
real-world context.
Order-of-operations rules must still be learned, but only for inserting
these operations into a calculator or computer program, since these machines
cannot visualize real world situations as a means of interpreting the symbols.

**Symbolization. **In standard mathematics, symbolization generally
means using symbols to represent numerical quantities. Even in word problems, students only
symbolize numerical values.
Symbolization in CIMM could not be more different. In CIMM, symbols are used to refer to
quantities, numerals and units.
Symbolization also means that every symbol has a definite referent; and
the student is able to either provide an example of that referent or point to
that referent, including symbols referring to operations. Students understand that numbers are
abstract-categorical symbols for relationships; and depending upon the context,
they are able to specify exactly what relationship the number is
referencing. This emphasis on
symbolization is an essential step in allowing student to develop and express
their reasoning processes.

**Manipulative skills. **Standard textbooks rely extensively on memorization
of algorithmic rules for manipulating numerical symbols. Most of the rules have little to do
with mathematics understood as a system for reasoning about relationships
between objects that involve quantity and dimension. Rather, they encode a
process by which the correct symbol can be magically constructed; e.g., two
minuses turn a parenthesis into a plus, so -(-6) becomes +6. Thus students who have a good ability
to memorize are successful and those who don’t become collateral damage
to an obsolete curriculum. The
memorization load can be greatly eased by changing the definition of what it means
to know and understand mathematics.

**Problem solving.** Most standard approaches to mathematics instruction
emphasize learning mathematics through problem solving. While mathematical
instruction necessarily involves problem solving, most programs rely on it
almost exclusively. CIMM by
contrast approaches problem solving as the application of mathematical ideas to
context-based situations where the focus is on learning how to apply the math,
not on learning the math itself.
This shift is possible because in CIMM the math is learned within a
generalizable context.

The unprecedented improvement in results in mathematics education being achieved with CIMM can best be understood if it is seen as a paradigm shift. The shift is away from seeing mathematics as a rigorous application of a set of axioms to a system built on observable properties of quantities and relationships among quantities. Thus the reasoning required in problem solving is driven by both context and axioms. The shift also allows for a student-centered, discovery-based mode of learning previously seen in other subjects but never previously successfully implemented in mathematics.

** **

** **

**CIMM implementation **

Implementation of CIMM requires commitment to a change in curriculum, pedagogy, and expectations of student capabilities. A teacher is required to take a two-week workshop (at minimum) to become fluent with the material. In addition, a coach is required for the first year to facilitate implementation of the program in the classroom and to develop the pedagogy of active engagement, including Socratic questioning and ‘whiteboarding’ presentation skills[1]. The initial focus on conceptual and linguistic development enhances the rate at which the material can be mastered.

Results of the CIMM program:

**Enthusiasm for and love of
mathematics**

A major impact of this program is the enthusiasm for and love of mathematics that it engenders in students and teachers. It is not uncommon for a student’s grade in mathematics to leap from a “D” to an “A”. This enthusiasm in turn translates into teacher enjoyment and satisfaction. In addition, classes utilizing the CIMM program experience a high level of student engagement.

**Improved test scores**

CIMM involves much more than content that is in a standard curriculum. However, all of the evaluations test standard performance objectives in mathematics. Where there is a low overlap between CIMM (CIMM requires a deep understanding of number sense and most remedial classes have very low number sense) and grade level curriculum requirements, there is little difference between classes taking CIMM and regular classes on district or state exams. However, at those grade levels where there is a high correlation, exam results can be as high as 13% over those normally attained.

** **

It is widely acknowledged[2] that the United States is lagging behind most industrial countries with respect to how prepared students are to enter fields associated with engineering, science, technology and mathematics. CIMM is a program that has the potential to raise the performance of U.S. students to the highest level in the world - and to leave the rest of the world behind. It also has the potential to virtually eliminate high school dropouts, turn around failing schools and greatly enhance student creativity and critical thinking in every area of school and life.

One obvious result would be an increasing number of students going on into careers in engineering, science and technology – and teaching. The shortage of math and science teachers is growing; and, as math knowledge standards for elementary teachers are raised, a shortage of teachers at that level can also be expected to develop. The love that teachers at all levels have for teaching this program and their widely-reported sense of understanding of what they are teaching will help retain teachers and attract new teachers.

The CIMM program is a technology innovation in the field of education and will itself become a valuable item in international trade.

Scientists educated in CIMM can be anticipated to see the workings of nature through a different mental prism and to make startling new discoveries. To paraphrase Seneca[3], future generations will wonder why these were not as plain to us now as they will seem to them. Discoveries with respect to the operations of the mind are no less spectacular than those of the skies.

February 2009

[1]
For a discussion of “whiteboarding”
see “Whiteboarding your way to great student discussions: Using large
whiteboards to facilitate group learning helps students communicate their
ideas”,* *David Henry, Julie
Henry, Stephanie Riddoch, Buffalo State University Publications, 2003,

http://physicsed.buffalostate.edu/pubs/Elem_whiteboarding.pdf,
and “Whiteboarding in the Classroom”, Dan MacIsaac, Buffalo State
College Publications, 2002, http://physicsed.buffalostate.edu/AZTEC/BP_WB/index.html

[2] Programme for International Student Assessment (PISA)
(Organisation for Economic Co-operation and Development (OECD)), 2006, pp.
51-54, http://www.pisa.oecd.org/dataoecd/15/13/39725224.pdf,
shows that U.S. student math score averages ranked 35^{th} out of 57
countries, while in the proportion of U.S. test-takers in the top two
performance categories (out of seven) the U.S. ranked 32^{nd}, and in
the proportion of test-takers in the bottom two categories it ranked 36^{th}
from lowest.

[3] Seneca, *Natural
Questions*, Book 7, first century,
quoted in Carl Sagan's *Cosmos*,
Random House, New York, 1980, p. xi: “The time will come when diligent
research over long periods will bring to light things which now lie hidden.
A single lifetime, even though entirely devoted to the sky, would not be
enough for the investigation of so vast a subject . . . And so this knowledge
will be unfolded only through long successive ages. There will come a time when
our descendants will be amazed that we did not know things that are so plain to
them . . . Many discoveries are reserved for ages still to come, when memory of
us will have been effaced. Our universe is a sorry little affair unless
it has in it something for every age to investigate . . . Nature does not
reveal her mysteries once and for all.”