Linkages

A word we use in the language field is "linkage."  This is the connection between things like the sound and symbol correspondence.  In math it might mean the connection between the quantity, the numeral and the name. 

Numeracy is such an essential component of all mathematics and it must be addressed if it is a deficit for any student.  Quantity awareness allows a student to fluently calculate, to estimate and apply.  It has a spatial component when thought of along a number line.  This can be an important component which lends itself to gross motor activities for the learning disabled child or the alternative learner. 

One aspect of numeracy that goes beyond subitizing is pattern recognition.  Think of place value and recognizing of four and four hundred are related.  Subitizing allows us to recognize quantity but pattern recognition allows us to apply it at higher levels.

The What Works Clearinghouse  suggests that children need to see multiple representations of math concepts.  Absolutely, I agree.  Keep in mind though that children need to have those representations linked so that the broad concept makes sense.  Think fractions, decimals and percent!   If a child comprehends 1/2 and taking one half of a quantity, he should also be able to link that to multiplication by 0.5 and taking 50% of a quantity.  The careful teacher makes sure to revisit previously taught concepts and connect the dots and not just teach each new topic as if it exists as a set of procedures unto itself. 

I routinely encounter teachers who attend my workshops and say, "I wish I had been taught this way."  I believe it is because I stress a concept based approach and as we would say in the field of dyslexia, an approach that is incremental, sequential, cumulative and thorough with practice to mastery.  To too many people though that sounds like procedures.  It is not.  The concept in math is the central piece for understanding.  Without, applications are hit and miss.  Procedural knowledge may get a student through one high stakes test but may be lost over time and not lend itself to applications such as problem solving.  Procedures do not lead to deep thinking about mathematics.

The NCTM has a new publication which is really worth a look.  Check out Principles to Actions on the NCTM website:   http://www.nctm.org/principlestoactions/