I am devoting my time, energy and working life to changing the “school” view of multiplication and multiplication facts. WE have DUMBED IT DOWN so far our students can barely muster the energy to remember or recognize that 2 x 5 equals 10. WHY?? Why in this ever-changing, ever growing incredibly complex and exciting world we live in have we chosen to turn our backs on making sense of mathematics in ways that engage students?
These are the responses of Grade 4 and 5 students when asked to explain what multiplication is. They indicate almost no comprehension whatsoever… and more disconcerting the teachers who read them were not shocked and disturbed…
MULTIPLICATION IS much more than JUST REPEATED ADDITION. You cannot use this thinking to understand the multiplication of decimals or fractions. What do you add to solve 0.45 x 0.67?
(0.45 + ?????)
Multiplication is dimensional. It grows by factors or units. It shrinks by factors or units. Students need to understand how to think in composite units. A simple idea like 3 x 1 is not really simple at all. Three times one means I have 3 singles but I am calling them one set or unit so I need to think in threes, not in 1,2,3.
Now three is manageable to the brain… It is small enough to subitize, to hold as an idea without counting. But how in the heck will I hold 8 as a composite unit. Eight is a huge quantity when my I look at a set of individual objects.
Area models, based on organizing EQUAL units into rectangular arrays illustrate the dimensionality of multiplication. Area models, based on organizing into rectangular arrays organize the count into a more manageable chunk.
CUISENAIRE RODS represent individual quantities as units. When I built with rods I am thinking about increasing by a unit, not by one. The brilliant colours and solid, smooth feel of the wood focus my attention. The rods translate to squared cm which allow me to develop my understandings of area and of metric measures at the same time that I absorb multiplication imagery into my long term memory.
Students need many many exposures to the idea of multiplication and how to develop relationships that are multiplicative… MEMORIZING sets of isolated, unrelated facts just doesn’t cut it. ‘KNOWING” IS MUCH RICHER THAN JUST MEMORIZING AND RE-GURGITATING….Listen to Dr James Tanton, 10 years as a research mathematician with a PhD from Princeton, 10 years teaching high school mathematics:
I memorize nothing… don’t get me wrong. I KNOW all kinds of math….
Generative Practice engages the body and the mind…. it requires thinking and deliberate mindful action on the part of the learner… It is practice that PROMPTS AND PROMOTES NUMBER SENSE.
BELOW YOU see a multiplication fact embedded in a hundred grid. One of my first principles: NUMBER IS IN NUMBER. The grid helps promote quantity recognition without reverting to counting by ones. The grid stimulates visual spatial reasoning….. “facts” are embedded in each other. The dark black referent lines help your eye see the 5s and use them to avoid counting. I can quickly see 8 across and 7 down.
Number Properties are a BASIC FACT our students need to make sense of.
The Commutative and Distributive Properties are the reason we can build personal strategies for recalling and relating multiplication and division facts.
The distributive property is named in outcomes at the Grade 4 and 5 level in the Alberta Program of Studies for Mathematics.
The Distributive Property makes interacting with the Multiplication Facts “playful”. Puzzles engage the brain and engagement is our goal.
Students practice facts in ways that connect visual spatial imagery to symbolic representations making the likelihood of long term retention and pain free recall more likely.
Can you see 8 x 7 in blue?
I see (5+3) (5+2) = 8 x 7.
(5+3) ((5+2) = (5×5) + (3×5) + (5×2) + (3×2)
(5×5) + (3×5) + (5×2) + (3×2) = 8 x 7
The distributive property allows me to see facts embedded in facts.
IT IS ALSO AN INTEGRAL PART OF UNDERSTANDING FACTORING.
Many Junior High teachers tell students a rule called F.O.I.L.
The problem with using F.O.I.L. is that you ignore or even negate the commutative property by insisting students memorize a procedure. Meaning is ignored. When students understand the Distributive and Commutative Properties , many can visually and mentally distribute without referring to any set of steps. They understand multiplication can be performed in any order and they gain fluency with manipulating facts.
OUR GOAL IS MENTAL FLUENCY WITH COMPUTATIONS.
THE GOAL IS UNDERSTANDING HOW NUMBERS ARE RELATED. THAT UNDERSTANDING FREES YOU TO THINK.
Flexible thinking is not just contained to mathematics. Flexible thinking builds confidence in one’s personal ability to solve problems without waiting for someone to remind you of “the way”. If the answer matters, I can find it in my head and I can tell you why it is the answer. That is more important that simply “finding answers.”
9 x 8 can be distributed as (5×5) +(4×5) +(5×3) + (4×3).
9 x 8 = (5×5) +(4×5) +(5×3) + (4×3) OR I can shorten that to 9 x 8 = (5+4)(5+3).
But this is just a starting point. What if you forget the referent lines that are pushing you to see and break each number at 5. How many other ways could you distribute 9 x 8?
What if I decomposed the 9 into 2 + 7 and the 8 into 2 + 6?
Can you build it? Can you explain it? Can you diagram it? Can you create the set of equations that connect to it?