Conceptual Math - Building Fluency and Number Sense

In 2019 I published a book on what I refer to as Conceptual Math.  This approach to learning math is to me the equivalent of phonics in reading, or learning the alphabet for writing.  I still think it is the greatest thing I have yet created.  

Here is a bit from the book on Numeracy:

​Patterns, Fluency and Number Sense
 
Dr. Daniel Ansari has been a strong supporter of Mind, Brain and Education, serving as the president of their International society. He and his team of researchers have illuminated much of what we know about moving from quantity awareness to the symbol. To cut to the chase, it is a developmental process.  One of my favorite studies looks at the relationship between high school math test scores and simple mental math.
 
He puts it this way “Arithmetic fluency, the speed and efficiently with which correct solutions to numerical computations are generated is thought to represent a scaffold upon which higher-level mathematical skills are built.”  (Price, Mazzocco & Ansari, 2013).  Basically lower-level number sense is important for higher-level mathematical success.  Now is that really surprising?
 
But here is the caveat.  The way the most successful math performers solved the task, was through utilizing brain regions that were not about quantity processing.  What we need to remember in math education is that numbers are not just amounts.  Fluency with numbers is in part moving beyond the simple notion of amount, and I believe into understanding relationship of the numbers within the base 10 system. 
 
The focus our teaching of numbers is traditionally on amounts.  We use concrete manipulatives and children count the little red chips, or dots on a page.  Research shows working with concrete examples and manipulatives helps learning.  I agree, we must have something in the physical, visual and embodied domains in early learning.  But numbers represent much more than amounts.  They represent distances, time, relationships, and it is good to work with numbers in a wider array of experiences than simply amount.
 
In fact most of math in science and engineering deals with relationships between numbers, how various things interact and impact each other as moveable parts of a whole.  It is limiting to a child’s active mathematical mind to focus so heavily on amount. 
 
There is a mathematical mind. Ask any mathematician.  It is highly connected to visual imagination and when mathematicians see formulae they see the whole array of what the numbers represent played out in their minds eye.  Some even describe equations as beautiful…or ugly.
 
We have created many varieties of manipulatives to help students understand tens, hundreds, and thousands in a more concrete way.  The small plastic blocks that come in ones, rows of ten, squares of 100 (10X10) and blocks of 1,000.  These are meant to make the experience more concrete. This has some value.  Part of our natural mathematical ability relates to what researchers refer to as our Approximate Number System (ANS). Training in assessing general amounts and discriminating between them has been shown to improve math ability, particularly in arithmetic (see Park, & Brannon, 2013). But there is considerable debate about the relationship between this system and the symbolic system.  Overall, there is a great deal of overlap in the activation of the symbolic number system and ANS, but there are also distinctions.  The symbolic system appears to be more lateralized into the left hemisphere, and the ANS has a unique area of activation in the right hemisphere (see Sokolowski, Fias, Ononye & Ansari, 2017).
 
The brain is a network of neural connections.  When we activate one region, its activation initiates connected regions.  This is necessarily true. What I propose here is that the general approximation system is doing more than calculating amounts.  If that were so, it wouldn’t be sensitive to the spatial arrangement of numbers, distance or size (Leibovich, Kadhim & Ansari, 2017).
 
All of this is to say, that we can move beyond the use of quantity in our teaching of number sense. 
 
One of the findings that has been significant to research in children’s number sense is that they are able to make relative comparisons to a number.  Using a scale, and relating the distance from one side to another, helped students see the relationship between numbers. 
 
This is important as it is not just amount, but the relative position of one number versus another to an anchor.  In our case, using the base then system, the anchor is 10, or any multiple thereof.
 
Conceptual math is all about number sense, understanding the base 10 system, and math fluency.  It is all about building a foundation.  It is about building a network of numeric relationships in the brain.  It is not a trick or strategy, it is a pattern generation system designed to activate our brains natural love of patterns.  ​