Post # 38 Multiplication, Division & Gozinto

“Many of my students struggle with organization, and I believe this is a major barrier to them retaining concepts and procedures. My algebra student, for example, consistently mixes up rules for adding terms versus multiplying terms, and my 6th grade student continues to forget the different rules for decimal operations. I have asked them both to make "Trouble Area" pages, and we've also done compare and contrast sheets, but it still has not been effective in helping them continue to add to it other difficult concepts, easily refer back to it, or ultimately master the material.”

This is a very good question.  It is a question that has a multitude of answers though.  Answers depend on a diagnostic prescriptive observation and approach.

When we think of applying rules to calculations such as the example above in algebra, we must consider the roots of this problem which go back many years.  The problem can be remediated through language, through simultaneous processing and guiding the student through appropriate language for an internal monologue.  The student who is confusing multiplication and addition with like terms and variable may not have fully understood the meaning of the distributive property with whole numbers. This begins in fourth grade with multi-digit multiplication.  It is knowing that we arrive at partial products which then must be added to attain the complete product.   Ex.  7(32) can the thought of as 7(30 + 2).  7(30) must be linked to 7(3) in the student’s mind and then adapted for place value.  Then the student must understand that he is multiplying “parts” of the quantity 32 separately and to get the total product he must add the partial products. Thus 7*30 is 210 + 7*2 which equals 14.  The total product is 210 + 14 or 224.  

Taking the student back to the box method of multiplication with simple whole numbers is a good way to make that linkage.  Then, after several one digit by two digit applications AND one digit by three digit applications, all using the box, the instructor can adapt to using x(x+3).  The teacher can use the same box method again but demonstrate how and why the distributive property works.  One of the problems with using FOIL with students who have never performed the distributive property with whole numbers is that they do not understand the place value implications of what they are doing.  It is not linked to anything the KNOW and so it is like a new skill coming out of nowhere…a skill which involves directionality challenges and rules based on words. 

Another challenge is that students do the calculations in their heads without language to support it.  They see 2 and 3 but do not register the operational sign.  Think back to “touch/say the sign, follow the line…”   This begins at a very basic level.  Getting students to note the operational sign is basic.  They must say the operation in order to perform the correct one. 

Another tactic is to say the sign of a term as its “first name.”  Thus the term 3x in 5(4-3x) is read 5 times (4 minus 3x); but for distribution it must be read as 5 times 4 and 5 times (-3x).  Its first name is “negative.”   This requires explicit instruction and practice.

Think what happens when we multiply (x-3)(x² +2x -1).  With FOIL the partial products are: x³ + 2x² –x –3x² -6x +3.  Now the student must add like terms that are not adjacent and recognize exponents as indicating place values.  With the box method, whose values are explicit on the diagonals. 

Now you apply language strategies to help with processing: On the outside we MULTIPLY but inside – the partial products- we ADD!  Chant it with rhythm, touch if need be, practice combining terms on the diagonals. 

The most difficult procedure in this unit is adding and subtracting polynomials. One method is circles and diamonds and squares “Oh My.” Circles and diamonds and squares!   Code like terms by surrounding them with shapes,  then add or subtract as indicated.  If they are written as two parenthetical expressions, use chanting again as a sub-skill:  (…..) + (….) Addition, remove (parentheses) and add.   (….) – (….) Subtraction, distribute (the negative) and add.

Again, explicit modeling and observed practice with sub-skills is important.  Only then can one re-integrate the sub-skills into multi-step equations.   Reduce the language.  Chant repetitive guiding language in simultaneous processing activities.  Add questioning to slow the student down.  Ask the student to reason aloud while processing until the error is remediated. 

Often at this level, the problems can be myriad: directionality, impulsivity, lack of internal self-monitoring, lack of internal monologue as moderator, lack of really processing the operation or operational sign, not touching the numbers or terms with a pencil point- (I call it air calculation), font that is too small to assist in visual processing and sustaining visual attention, gaps in conceptual or procedural knowledge….etc.

A Few Remediation Strategies: 

·       “when in doubt, write it out”

·       Font too small- blow it up, use dry erase and super-size it.

·       Color code

·       Touch and say – the pencil point focuses the eye, the voice slows down the speed, the simultaneous interaction of senses is more apt to catch errors.

·       Chant the rules

·       Note the sign  

·       Link to previous skills, real life or whole number operations then apply to the abstract.

·       Visual dictionaries with examples to codify rules – Major concept sheets rather than single operation sheets.

·       Consistency in concepts, vocabulary and many operational procedures from early skills to algebra- Using similar formats and linking them in instruction