Christopher Woodin, Ed.M.
His training is in neurology and teaching.
Landmark School
Manchester, Ma
Also available: handout, web site (http://woodinmath.tripod.com)
Effective Strategies to teach multiplication and division facts using whole
to part, visually-based graphic organizers. A standardized diagramming system
prompts integration of facts with related word problems. This framework provides
a means to solve word problems, then students compose a problem using the inverse
operation.
This is research based (dual-coding theory). If you label something, you will
not necessarily have a visual image. But if you image it, the mind naturally
adds a word.
Whole to part processing is that which helps the students.
Our verbal sphere is organized by semantics, not by function. So when we learn
facts they don’t have a picture unless we give that to them.
Instead of giving the students the number, give them the picture and let them
come up with the number.
Give the sum first and have the student come up with the sentence(s).
Using the Gestalt allows an organization for the information.
Word Problems: Highlighting just becomes a highlighted word problem. Don’t
highlight until after the problem is done, reflecting then on what the keys
words were.
Linear diagrams allow students to see the problem better. Put all the information
into a box and then use the linear diagrams to drive the solution process.
They can then cover up the part they know, and see what piece is left.
Then have the students make up an inverse word problem to solve. Doing lots
of problems will not help as well as rewriting each word problem and converting
it.
Multiplication and Division are taught simultaneously. The box used for multiplication
can (with two lines erased) shows the division process.
Math Facts Use a 1-100 chart, (see notes)
Emphasize the visual image, the Arabic numbers and then the word. If the child
is doing part to whole in their mind (7 x 2 = 14) there is a lot of language
and chance of confusion.
Many kids have a good visual memory, but there planning is off. Use hands to
show the five finger and two fingers. Picture your fingers and then your fingers
again. 14 = 7 x 2
Again, use the sum first.
Do an activity in the class, then double time it. Do the concrete example always.
Multiplication Tables: Make a geoboard and make it the size of a regular inch
paper. (then unifix cubes will fit in) Student can drive nails in as well, but
then write in the actual numbers based upon what they find out.
Word Problems: Associate specific nouns with the fact family being learned.
Group names : Subordinate nouns (elements)
Single step problems can be made visual with boxes. The boxes will show the
students how to break the problem apart.
Students can then highlight the component structure of a problem.
Be divergent with the information. Don’t just isolate the key word until
after the problem has been diagrammed.
When teaching teach addition/ subtraction and multiplication/division. Must
teach in parts as the language can all get confusing. Doing multiplication/
division you are looking for groups. One piece of the boxes always has to be
there.
1-100 chart – highlight 5 down and 10 down in a combo 2’s and 5’s
color (because 10’s have the same factors)
5’s – always teach with nickels. (If you click nickels they make
a distinctive sound) [also if you rub two nickels together quickly it drives
squirrels nuts J] Always use iconic way of representing 5.
Time/ Clocks – uses a circle on the floor
Teach the fives by having the students
identify the 3, 6, 9 and put that on graphic organizers to visualize and see
the 5 time facts. This is a way to visualize time. Many LD kids have a poor
concept of time and this is a way to make it concrete.
Multiplication Table: Add the ones (gray) and teach multi-digit multiplication
with known facts.
9’s – all digits add to be a multiple of 9. the worst thing to do
teaching the facts is giving the mad minutes when they are not ready. It cements
the wrong things.
Do the boxes again. Nine facts are one decade less than the 10’s. Also
use the diagonals on the 1-100 chart (see notes)
Perfect squares: use 100 centimeter graph paper. Make sure they come up with
the number. Efficient memory retrieval has to do with efficient memory storage.
2,3,4,5,6,7,8,9 (4,9,16,25,36,49,64,81)
6’s – Even multiples of 6 have one’s place digit that match
their 6x factors.
7’s – or any facts. They are all a combination of other factors.
Student says, what is 7 x 7 ? Write it down. Then label one factor with a common
noun. (i.e. 7 weeks) then show me seven fingers. Label the fingers as weeks.
What is 2 times 7? (14) What is 5 times 7? (35).
The ladder chart for 8. – see handout.
Impact:
This tool for doing word problems makes great sense. Also the relational way
of showing the math facts really helps. Altering syntax to make things whole
to part really helps. The means of explaining the process is great. The handout
is excellent.