“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