Download Myths Workshop 2015

Myths & Lies In
Teaching Mathematics
or
Do You Believe in
Unicorns?
From Comprehending Math by
Hyde
From a Grade ½ Classroom: A young man raised his hand
with a simple inquiry. “Why, Mrs. Buddy,” he said, tongue
thrusting wildly through the space where front teeth are
usually found, “why do you call it predicting when we’re
talking about reading, hypothesizing [which came out as
hypothethithing] when we’re in science, and estimating
[ethimathing] when we’re in math? Aren’t they really all the
same thing?”
Learning Goal
Participants will leave with greater awareness of the
need for consistent precision in the teaching of
mathematics so that the Big Ideas with “long legs”
(Concepts and Processes) are more effectively taught
across grades.
Success Criteria
Not co-created – it’s an hour, live with it
Math Learning Goals and Success Criteria
come out in the Consolidation NOT the
Minds On.
Some cognitive dissonance
In classrooms (or in workshops you do) the focus
shifts to Big Ideas to create more precision and
elimination of “short cuts” in math
Agenda
1.
Why do we lie to students?
Minds ON: Non-Math Cultural Lies & Language Lies
that effect math
Action
1.
2.
3.
“Pure math” lies
Culturally held myths about math – larger community.
Assessment Myths based on wrong-headed math
Consolidation
Pedagogically held myths
Myth or Lie
Lies in math: statements/teachings which are
mathematically incorrect
e.g. “You cannot subtract a larger number from a
smaller number”
Myths teach deep, usually unquestioned, culturally
beliefs. The telling of these myths conveys the belief in
an emotional tone that creates great affinity.
Often done through narrative.
e.g. “Mathematics is about getting the single correct
answer.”
Santa Claus vs. Damage
The story of M – no Santa but lots of evidence
What is the intent of this lie to primary students: “You
always subtract the smaller number from the larger
number.”
What other reasons cause us to lie to students in math
class?
Minds On: Cultural Myths
There is no such thing as:
Vegetables
Sometimes we eat roots/tubers, stems/trunks, leaves, nuts, berries, fruits
Common sense
Just ideas held in common by a culture or subculture which may or frequently don’t
apply outside that situation (Grub)
Academics
Usually means we did it with paper/traditional educational methods only
Standards
The only true standard in education is each student achieving everything they
individually can
Creativity
An all inclusive term for originality, making connections, using a different medium,
combining beliefs from different cultures, …
Creation is a verb not a noun – a process
Liars,
Damn Liars,
Teachers
(With apologies to Disraeli)
On the sheet you have a compilation of mathematical
lies that are sometimes (often?) told to students
With a partner(s): read them and discuss why they are
lies
Rank your “worstest” lies (3): those that would do the
most long term damage
Note: the absolute most damaging lie isn’t on the list –
I will do that afterwards
Lies Told In Language Class
That Affect Mathematics
The topic sentence of a paragraph is always the first sentence.
Key words help to “decode” math problems
Tale of Two Cities full version: …
Student Edition: Best time and worst times. Must be an addition problem
Don’t teach kids to NOT READ.
We process everything through our language
Other language myths that are non-mathematical but proven through
mathematical analysis of linguistics:
Grammar rules are static and universal
English is a phonetically based language
Weekly spelling lists improve spelling.
BUT THE BIGGEST MYTH IS:
What did you decide?
Top 3
Are there any you have a question about why it is a lie?
When you subtract the answer gets
smaller / you always subtract the
smaller number from the larger
number
5 – (-3) =8
Very young children understand integers (next slide)
Kylie is in
Grade 1
Borrowing and Carrying
In our number system you cannot actually put more
than one digit in a value place
What’s actually going on conceptually …
Big Idea: What is Subtraction?
Take Away 5-3 = 2
Difference Between
5 + (-3) = 2
Black = +ve, Orange = -ve
The additive inverse to
zero (the basic
operation of integers)
+1 + (-1) = 0
Big Idea: Adding zero does not
change the number!
We can extend the number of value places we put
zeroes after the decimal, which indicates precision or
need – but it does not change the value of the number
Big Idea: Rational Numbers
Counting Numbers: 1,2,3, …
Whole Numbers: 0, 1, 2, 3, …
Integers: … -3, -2, -1, 0, 1, 2, 3, …
Rational numbers: (fractions)
It is the relationship between the numerator and
denominator that is central not the actual numbers:
One half is equivalent to two fourths to three sixths etc.
Big Idea: Rational Numbers
You always divide the smaller number into the larger
number
When you divide the answer always gets smaller
Big Idea: The Decimal Cannot
Move & Has No Number
Value
Lie: the decimal moves when you multiply by 10’s.
Big Idea: multiplication & division
represent multiplicative thinking the basis
of all proportional reasoning
Addition and subtraction are linear operations: we
always represent on a number line
+7
+5
0
5
12
Multiplication and it’s inverse division define
proportional relationships (Big Idea: Rational
Numbers)
32 x 12 = 384
12 x 312 = 384
384 ÷ 12 = 32
384 ÷ 32 = 12
32/12 = 2.666666 long side/short
side proportional relationship
12/32 = 0.375 short side to long
side
Big Idea: there are many,
many ways to perform
operations
These may be cultural, traditional/historical, based on using
less paper, less space
They should be based on: what makes it easier based on the
actual numbers in the question: “Look to the numbers!”
It may help to line up decimals with the North American
traditional addition/subtraction algorithms but is of no
value for multiplication or for other +/- methods
Big Idea: If it gets them there,
don’t stop them …
… as the numbers get bigger they will begin to choose
when fingers work
Astrophysics
Big Idea: We actually teach 3
number systems:
Our Base 10 number system which is multiplicative at it’s
heart
Time measurement systems: sometimes based on 60,
sometimes 24, sometimes 365, sometimes 12 (wonder why
this is so confusing for kids to start out???)
Money: based on 100 not 10 but is close enough to be
confusing (units of 1 (or used to), 5, 25, for partials and 5,
10 etc. for wholes
or
is said one fourth, it is not a quarter hour
or an actual quarter!
Big Idea: part of quantity is to
have students read numbers
with proper mathematical
precision (Place Value)
27.56 is read:
Twenty-seven (two tens and seven ones) AND fifty-six
hundredths (56 over 100)
Common parlance is not acceptable to teach the precision
of mathematics: this will make a huge difference in your
student’s understanding of decimals and fractions (all
partials) and their proportional reasoning.
Big Idea: Fractions are
represented using 3
concrete/visual models
Never use the pizza model because of lack of precision
Because we always use the area model – we leave out
the others:
Our problem choice needs to use them all – in balance
“Conceptual understandings lie in the multiplicity of
the representations”
We use linear and set more frequently than area in real
life
Set Models
In a set model, a collection of objects represents
the whole amount. Subsets of the whole make
up the fractional parts.
1
4
Area Models
In an area model, one shape or object represents the
whole. The whole is divided into fractional parts.
Use colour tiles. How many ways can you model 1 of 12?
4
Linear Models
In a linear model, a length is divided into fractional parts.
0
1
4
1
2
3
4
1
Use coloured relational rods. How many ways can you
model ?
1
4
Big Idea: if fractions
(rationals) are a relationship …
… isn’t the relationship still the same whether it is in
non-lowest terms or decimals or improper fractions (I
love the pejorative of “improper” fractions – so
Victorian!
The form for the fraction depends on what I want to
do next with it!!!
Big Idea: the accuracy of the
measurement is determined by
the context
Building a deck – eighths of an inch
Distance to a near star: billions of km
Bank account: depends on your overdraft limit!
Big Idea: area is measured by
iterated congruent square
units
The formula (short cut) to find the area of a rectangle
is:
Arec = l (long side) x w (short side)
The formula for the area of other shapes
are NOT
Big Idea: = (this symbol) states
equivalency (not the same)
If we spent time in Grade One (ever) and in other
grades filling out sheets that look like 5 + 7 =
If in grades 7 and up we say do the same thing to the
left side as the right side this is not correct:
Perform an equivalent operation
We teach:
The answer goes to the right (not equivalency)
We do long term damage to the child’s development in
algebra
Big Idea: Algebra is the
representation of patterns
Sometimes we use variables, constants and coefficients
Sometimes we use tables
Sometimes we use graphs
Formal algebraic representation allows us to explore
amazing things about math and allows us to predict
which is why it is so important and powerful
But all of it is just a pattern!
The Biggest Lie of All is:
Drum Roll please ….
The Long Division
Algorithm
"If you have done two long divisions in
your life, you have done one too many!"
Gaspard Monge` (Father of Differential Geometry.1746-1818)
Long Division Lies
Place Value, Place Value, Place Value (Why do we start
there!?
What is a “Gazinta”? (Units of is the proper term.)
What is a “Bring Down”?
The Myth of the Standard
Algorithm
Ethnocentrism perhaps even racism
The Doubling and Halving (binary) ancient algorithm story
The reason we study algorithms has changed
Should we still teach the two-digit multiplication algorithm?
The algorithm is the operation
The conceptual understanding lies in the multiplicity of
representations: concrete, visual, student created,
traditional/cultural methods, algebraic, digital …. Not in the
ability to use standard algorithms
Culturally Held Myths About
Math
Usually a result of traditional methods of teaching
math that lacks the interactivity and problem solving
and so creates unknowing myths
Myths Educational Methods
Implicitly & Explicitly Created
Math is about the correct answer
Math is about the precise (singular) answer (an estimate that fulfills
the question is not acceptable.)
Mathematics = arithmetic
Math is about speed and efficiency not elegance and beauty
Creating stress
Math is about the method with the least steps (disguised as most
efficient)
I will never use: < Insert Topic Here>
Use of calculators decreases operational skills
Larger Myths About Math
Math Ability
“I cannot do math and it is genetic”
(This is actually the only thing that the amateur John Mighton
gets correct)
Math is Acultural
Math is universal
Math is about learning rules
If I memorize these rules I can do math
MATH IS EXTERNAL TO THE LEARNER:
anti-constructivism
Lies Told In Language Class
That Affect Mathematics
When we pretend that most math “word problems”
actually make sense:
How Old Is the Shepherd
Assessment Lies Based on
Algorithmic/Arithmetic
Understandings of Math Leads to:
Abdication of professional judgment in assessment (evaluation) to the mean (average)
Median or mode with adjustments for professional observations and conversations
Abdication of professional judgment in assessment (evaluation) to “mark book” or a mark
book
Is mark book set up to mode (as is required in the policy document “Growing Success”?)
Are the overall expectations weighted more than specifics? Which specifics?
Have you built in a “fudge” factor to allow you to adjust for professional judgment and
do you actively decide to use/not use it every time you provide it to students/parents
I can assess (evaluate) to a single number with honesty, accuracy, and integrity and I can justify
it to the overall expectations.
Assessment and evaluation is so complex even God only uses pass/fail.
Consolidation: Myths & Lies
Turn to an elbow partner and share an understanding
you constructed or further developed or re-confirmed
here. (Before I tell you what the really big Myths are)
Traditional Math Instruction Was Effective
Ricky Henderson
Math Phobia/Avoidance/Anxiety
Creating Stress
“If I have a PhD in math , I know how math should be
taught because look, I was successful, so I must know.”
Consolidation: Myths & Lies
The purpose of teaching math is to get students ready
for not just calculus, but differential calculus.
Math homework actually works (except for academic
secondary kids)
Strategies vs. Structures (next slide)
Strategies vs. Structures
Strategies:
(Proven effective)
Open task or parallel
task
Graphic organizers
Cooperative learning
Compare and contrast
questioning
Concept attainment
AfL/AaL
….
Structures:
(useful only if some if strategies
embedded)
3 part lesson plan
2 part lesson plan
centers
Feel Free to Use Any of These
Materials
Just give credit:
Dan Peter
“Teaching As Learning”
[email protected]
Some Links
Somebody made a Prezi with a few of these:
https://prezi.com/r-zashsuwb_2/9-mathematical-lies-iwas-told-in-elementary-school/
NCTM Article: “Mathematical Lies We Tell Our
Students”
http://www.jstor.org/stable/10.5951/teacchilmath.21.4.
0197
(You have to pay for it unless you have a subscription)