Thinking Vertically About Teaching Math

Thinking Vertically:  For remediation with older students, one needs to think of skills introduced but not mastered.  Begin to think about concepts which bridge multiple operations and levels.  Two examples are "Regrouping" and "Place Value."  With older students who have been taught procedures without concepts this is a terrific place to begin. 

Start by using manipulatives to model whole number operations with the place value mat you were given.  Ask students to "prove by construction" answers to basic problems without regrouping.  With severe students I would recommend the craft sticks because as I have said, the student may need to physically bundle and unbundle quantities.  If the student needs only to reinforce the concept, base ten blocks may be used.  After regrouping is introduced, practiced and mastered, you can move the student to fraction concepts.

Creation of fractions with your fraction circles is a first step.  Students may keep one circle uncut to remind them of how many pieces it takes to make a "whole."  Other fraction pieces may be used to add and subtract like fractions.  If the solution is an improper fraction, the students quickly see that laying the whole circle on top "simplifies" the fraction to a mixed number.

The next step is helping them understand that we may "regroup" from the one's place value to the fraction place value by moving "one" to the fraction place value and representing it as the required circle "cut" into the required size pieces.  The whole can never be in the fraction place value unless it has been "cut" into its required number of pieces, thus creating an improper fraction in the fraction place value.

The student learns that we may get a "sum" which is improper in any place value by the operation of addition.  We then "simplify" the quantity to its proper form.  We may need to create an improper quantity in ANY place value in order to subtract.  This is a fundamental concept for both whole number operations and fractions.

The older student feels validated in that he or she is working at higher levels of math, but is also beginning to understand the fundamental math concepts which form the foundations of higher level skills.

What Comes Before & What comes After

What Comes Before & What Comes After 

In thinking about the math courses, I wanted to ask each of you to consider thinking about one concept such as multiplication or fractions.  Begin to think about how that concept appears at various levels of instruction.  What would be the earliest exposure a student might have?  What vocabulary is essential for the child to comprehend the concept?  How could a child experience the concept, practice the concept and demonstrate proficiency at an early level?

Then, I would like you to jump ahead several levels and years.  How is this concept applied at higher levels of math?  How does the early vocabulary continue to be important in concept formation and application?  How does one expand this concept to extremely abstract levels?

As a primary grade teacher, we need to understand how what we do at basic levels forms the foundation of what is to come.  As a secondary teacher, we need to understand the basic concept instruction and vocabulary so that we may go back to fill in gaps for those who need remedial instruction.

You might also choose a concept such as division or multiplication.  Try to spend a few moments considering the various levels and applications