Myths & Lies In Teaching Mathematics or Do You Believe in Unicorns?
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Transcript of 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
Action1. “Pure math” lies
2. Culturally held myths about math – larger community.
3. Assessment Myths based on wrong-headed math
ConsolidationPedagogically 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 senseJust ideas held in common by a culture or subculture which may or frequently don’t apply outside that situation (Grub)
AcademicsUsually means we did it with paper/traditional educational methods only
StandardsThe only true standard in education is each student achieving everything they individually can
CreativityAn 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
MathematicsThe topic sentence of a paragraph is always the first sentence.
Key words help to “decode” math problemsTale 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:
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)
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
The additive inverse to zero (the basic operation of integers)
+1 + (-1) = 0
5 + (-3) = 2
Black = +ve, Orange = -ve
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 ValueLie: 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
Multiplication and it’s inverse division define proportional relationships (Big Idea: Rational Numbers)
0 125
+5 +7
32 x 12 = 38412 x 312 = 384384 ÷ 12 = 32384 ÷ 32 = 1232/12 = 2.666666 long side/short side proportional relationship12/32 = 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
is said one fourth, it is not a quarter hour or 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
27
Set ModelsIn a set model, a collection of objects represents the whole amount. Subsets of the whole make up the fractional parts.
14
28
Area ModelsIn 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
of 12?
14
29
Linear Models
In a linear model, a length is divided into fractional parts.
Use coloured relational rods. How many ways can you model ?
41
21
43 10
14
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 unitsThe 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
patternsSometimes 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 racismThe Doubling and Halving (binary) ancient algorithm story
The reason we study algorithms has changedShould we still teach the two-digit multiplication algorithm?
The algorithm is the operationThe 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 CreatedMath 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 rulesIf I memorize these rules I can do math
MATH IS EXTERNAL TO THE LEARNER: anti-constructivism
Lies Told In Language Class That Affect
MathematicsWhen 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 EffectiveRicky 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-i-was-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)