Math Learning “Underpinnings”


Barry Walker, Science Department Chair,

Briarwood Christian School, Birmingham, Alabama



I feel an urgency and importance for implementing Cognitive Instruction in Mathematical Modeling (CIMM or simply “The Dots”), because the majority of students cannot learn algorithmic skills without conceptual understanding. Current strategies are not working and change is necessary.


CIMM is a definite paradigm shift in what it means to know and understand mathematics.  Even though its worksheets and teacher notes are in early stages of development, far from Prentice-Hall or Houghton-Mifflin standards, they do highlight a new beginning.  CIMM teachers must “work” at adjusting and readjusting their plans every day to accommodate the understanding acquired by kids.  I am using CIMM in four “grade level” 11th and 12th grade classes. This is a lot of work but I can’t call it “work” because I’m seeing kids finding “The Pleasure of Finding Things Out.” I embrace this wonderful program because my kids are actually smiling in math class! CIMM and I are greatly enhancing and enriching kids’ math education.


Bonnie, my wife and partner, is a very bright artistic, visual and literary person. She carefully looked at the CIMM test my classes took and responded, “This is helpful for me because I can’t see ‘numbers’, but I can see and visualize ‘objects’ and these dots.”


As I reflected on her comments, the word underpinnings leaped into my mind.  I first heard this word this summer in a Modeling Physical Science teacher workshop at Arizona State University from my instructors, Lee Rodgers and Pat Burr.  In their 9th grade physical science classes of low SES students they spend a quarter or more a year just building the essential “underpinnings” so that a student can effectively understand and represent the science presented in the course.  In physical science the underpinnings are the metric system, measurement and estimation, length, area, volume, identifying relationships, experimental design, graphing, etc.  Since kids do not have much “number sense” about using numbers to represent relationships when they come into Physical Science, they even work on that “underpinning” by using “the dots”.


So, I’m forced to ask myself and American math education, ”What underpinnings are we giving our kids that will truly aid them in learning math and erase that phrase, ‘I hate math.’?”



From Free OnLine Dictionary:

un·der·pin·ning  (ndr-pnng)


1. Material or masonry used to support a structure, such as a wall.

2. A support or foundation. Often used in the plural.

3. Informal The human legs. Often used in the plural.

From Wikipedia:

In construction, underpinning is the process of strengthening and stabilizing the foundation of an existing building or other structure. Underpinning may be necessary for a variety of reasons:

If you just glanced over the previous descriptions, please go back and slowly look at each thing in relation to what we have given to at least 80% of American students as their “underpinnings” in mathematics.  They are living in a structure built on memorized mathematical algorithms. As more and more math is piled on their weak conceptual underpinnings, many are unsuccessful to the point where they sink down and simply say, “I hate math” and give up.  The cost of this is enormous for them personally and for our society.  I am able to reject the belief that this is the best we can do, simply because I can see another way! 


CIMM is about building “underpinnings” that can support the abstract symbolic world of mathematics.  A student’s world is changing rapidly and they just don’t have the underpinnings, “legs”, to mathematically stand tall in that change.  Purely memorized algorithms only work in the narrow context in which they were memorized.  As the number of mathematical algorithms grows they are easily forgotten because “meaning”, the “handle” that connects, is missing.  CIMM connects mathematical symbolism and algorithms to groupings, quantities and relationships that can be visualized, remembered and applied.  Mathematical algorithms are essential to using mathematics; they just need a “handle” called “meaning.”


Where is the best place to introduce CIMM into a child’s education?  I strongly suspect the elementary school is the place to start.  The earlier we start the less we have to clear aside before building strong underpinnings.  But old foundations can be replaced at any age or experience.  I’m 65 years old and have learned more useful math in the last three years than the previous 50!  Until CIMM is implemented in the elementary school we need to teach it anywhere possible in grades 7 through 12.


So, do we wait until CIMM is elegantly written and presented with “research numbers” that “proves” it really works?  Or, do we trust what we have seen and experienced so far, “our gut feeling”, and join in the development of these “underpinnings”?  You know my vote already.


Martin Luther King said,

            “You don’t have to see the whole staircase,

              Just take the first step.”