Concerned with a population of increasing girth, Brazil
recently came out with a new food guide. Two of the basic principles are; cook from
scratch and eat with others. These are
also basic principles in my classroom.
Much of what we do in the classroom is the intellectual
equivalent of fast food. Relying on
textbooks for example. I would rather my
students developed their understanding from scratch though activities. Activities are the learning equivalent of slow food. Research shows that eating alone
can lead to unhealthy eating habits. I
believe that learning math in a group setting promotes healthier thinking. The student desks in my class are arranged in
pods of 4 to promote collaborative learning and encourage accountable
talk. (I do not want to discount the
value of being alone with your thoughts when conceiving ideas but ultimately
those ideas must be shared, tested and scrutinized by peers - hence this blog.)
Consider the number of questions in a textbook. The 700 page Calculus and Vectors textbook that
I have in front of me has over 2000 questions.
It is safe to say that most of these are not meaty problems; more like small
bite-size snacks. Of course no one does
every question in a textbook. It is
assumed that you will pick and choose or simply 'ace it'; "Do questions 1
to 25 odd only; a, c, e....". Now, I
know there are great and interesting problems in textbooks but most of the books
tend to be filled with appetizers that are supposed to prepare your taste-buds
for the main course. Instead they tend
to fill you up, leaving you bloated and sleepy.
I think that after reading The Man of Numbers by Keith Devlin, I have a better sense of our textbooks' pedigree. It is an account of how Leonardo of Pisa, better known as Fibonacci, published The Book of Calculation in 1202. Early math textbooks were created to teach merchants commercial arithmetic skills. This pivotal textbook set the standard for
years to come. In fact, other than the
addition of web links, textbooks haven't changed much since. In an age where there are calculators in just
about every kid's pocket, maybe it is time to start question what we value in
Math class.
Historically the ability to do fast and accurate
calculations has always been rewarded.
Students needed to know their "number facts" to allow them to solve more complex questions without
being bogged down with basic
calculations. I get that. But is it so bad to use a calculator for
bigger numbers? Unfortunately the value
we place on snappy answers tends to permeate our work; question/answer wait
times averaging less than a second; mad minutes; unconnected one-mark questions
on evaluations; timed testing: rapid actions with limited capacity for nourishing
deeper thought.
I know that calculation skills are important, but let's not
be nostalgic for our glorious past of memorizing multiplication tables. When someone in the back-to-basics movement
complains that a young store clerk (a casualty of the 'new math') can't do
simple arithmetic in an electrical blackout, I think of an old-school office
administer who asked for my help with percentages. How to best develop basic skills is not the
main issue. A bigger concern is our reluctance
to even consider applying mathematics outside of the math classroom. Even people with solid computation skills may
never think of doing any math to see if an electronics store's extended service
plan makes sense or if locking into a 5-year natural gas plan can actually save
money. Do our classrooms nurture or destroy
math for most?
Another idea I have adopted from the slow food movement is 'eat
local'. If you close your textbook, you and
your students may start to see mathematics as way to understand the world
around us. For example, when a local car drove into a sinkhole, I used the story in my grade 9 class to dig into concepts of area, volume, ratio and
proportion. One of the students knew the make and model of the car. We found the car's dimensions online and
calculated the volume of the hole. We
built scale models of the hole using linking cubes and built a life-size model
with desks and plastic sheets. When the news reports stated that the hole had
grown to the size of an Olympic-sized swimming pool, we used rope to outline a
pool on the playing field. What does
two and half million litres look like?
But doesn't all that foolishness with ropes and cubes take time
away from the hard math that these kids will need for the next grade? There are so many curriculum expectations and
so little time. It seems we are always
in such a rush to cover the curriculum. To do that we need all the students to move
along at the same pace. It reminds me of
the story about the man riding a bus in Moscow.
He watched as the driver failed to stop and pick up people waiting at each
bus stop. The man finally asked the driver why
he didn't stop to pick up passengers.
The driver explained, "I am on a tight schedule and if I stopped I
would no longer be on time". We
need to pick up all the passengers, even if it seems to slow us down. But here is a trick. We don't need to get every student to the
same place at the same time. Moving a
group of toddlers down the street while they hold on to a rope is safe but it's
not efficient. Some will be bored with the pace and others will be tripping
over their shoe laces. I say, let the
faster students run ahead while we concentrate on helping the slower students.
We will all be further along - in the long run.
Is getting everyone through the curriculum on schedule always
effective? I recall a colleague's
account of an entire class of grade 11 academic students swearing that they had
never seen the Cosine Law before. They
all were very adamant until one of the students found the Cosine Law on a test
from the previous year; an evaluation on which she had done quite well. We often introduce a concept with a concrete
activity but quickly move to more abstract representations and concepts. Pushing
into the abstract too quickly tends to lead to limited understanding and outright
misconceptions. Many students cope by desperately
clinging to formulas. My grade 9 students
come to my class knowing that A = l x w but many need to count the squares one-by-one
in a 5 by 4 grid. They all know A = bh/2
but most never consider that a triangle has half the area of a rectangle. Similarly, students can recite "a
squared plus b squared equals c squared" and then tell me that 3 + 4 = 5,
or 42 + 52 = 32. I have had students tell me that they are
quite positive that a positive times a positive is a negative. And as every teacher knows that once
something is 'learned' it is extremely difficult to 'unlearn' it. Giving students a formula, rule or trick for
the sake of expedience will ensure that we all get to 'teach' basic concepts over
and over in grades 7, 8, 9, 10 and again
in grade 11 and even 12.
Slow down. Developing
concepts takes time. Making connections
takes time. Formulating good questions
takes time. To solve complex problems without having the teacher slice
everything into small chewable pieces takes time. To reflect on the validity and reasonableness
of a solution takes time. To build an
repertoire of complex skills takes time.
Slow down and in the end we will all be ahead.
The government of Brazil knows that fat-free, calorie-counting
diets won't stem the rising tide of obesity. Taking a page from the slow food movement, they are banking on a simple approach to save medical costs and
ultimately people's lives. Fast food is not cheap. It's nutritional value is minimal
and the overall negative effects on our health and the environment is
significant. Thinking that you are
saving yourself time and money with fast food is false economy. Thinking that you are saving yourself time
and effort in your classroom with snappy questions, tricks and shortcuts is
also false economy.
I am not the first to use a food analogy for education. Give a
man a fish and he will eat for a day, teach him to fish and he will eat for a
lifetime. The word fish acts as a noun and a verb. My attempt to rework it is not quite so nice; give students some strategies and they will
understand for test-time but teach students to strategize and they will create
their own understandings for a lifetime.
It's time to break from our diet of quick and easy questions,
formulas and tricks, the cognitive equivalent of salt, sugar and fat. Time to
construct mathematical concepts from scratch with the basic ingredients of investigation,
questioning and accountable talk.
Fast food is junk food.
Slow food is best. Fast education
is junk education. Slow education is
best.
