Background
Evil Forces
Role of Analogous Tasks and Examples
Slaying Misconceptions with Analogous
Tasks
Dr. Roger Fischer
EMAT Project Facilitator
Montana State University
August 29, 2015
Conclusion
Background
Evil Forces
Role of Analogous Tasks and Examples
O VERVIEW
Background
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
Conclusion
Background
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
M Y B ACKGROUND
I
B.A. Mathematics: Teaching Secondary
I
I
I
I
2004, Central Washington University
M.A.T. Mathematics Teaching
2006, Central Washington University
Ph.D. Mathematics: Mathematics Education Concentration
I
2014, Montana State University
I
Two years teaching middle school mathematics
I
One year teaching developmental math to undergraduates
I
Eight years teaching college level mathematics content and
pedagogy
I
Currently in my second year of teaching developmental
math to undergraduates
Background
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
T HE P ROBLEM : “E VIL F ORCES ”
Warm Up: Write the following expression in simplest form.
Then suggest some difficulties students may encounter as they
work.
2y2 − 4y y2 − 2y − 3
·
2y2 − 2 y2 − 5y + 6
Background
Evil Forces
Role of Analogous Tasks and Examples
E VIL F ORCES
One particularly robust misconception students have is
centered on the idea of “cancelling.”
Conclusion
Background
Evil Forces
Role of Analogous Tasks and Examples
T HE M ATHEMATICS
Let’s revisit a previous example, with the following
terminology in mind.
I
Factor (noun): A multiplier
I
Term (noun): A number, a variable, or a product or a
quotient of numbers and/or variables
Conclusion
Background
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
C OMBAT E VIL F ORCES WITH A NALOGOUS TASKS
Simplify the following expressions. Try to find two different
approaches.
12 + 8
12
5 + 40
5
What are some advantages and disadvantages of each
approach?
12 + 6
24 + 18
Background
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
A NALOGOUS TASKS
Write each of the following expressions so that the numerator
and denominator have no common factors.
12x + 8
12
5a + 40
5
3m2 + 3m
6m2 + 9m
Background
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
A NALOGOUS TASKS
Write each of the following expressions so that the numerator
and denominator have no common factors.
x2 − 25
x2 − 10x + 25
4a2
3a − 6
·
2
3a − 12a + 12
2a
3a3 + 3
6a2 − 6a + 6
Background
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
ATTENDING TO P RECISION
I
I
I
a
=1
a
This fact underlies our strategy for writing rational
expressions in simplest form.
For all non-zero a,
Without consistent use of precise vocabulary, “factor and
cancel common factors” becomes “cross out things that are
the same.”
Background
Evil Forces
Role of Analogous Tasks and Examples
A NOTHER C OMMON M ISCONCEPTION
Everyone’s favorite “identity:”
(a + b)2 = a2 + b2
Why is this misconception so robust?
Conclusion
Background
Evil Forces
Role of Analogous Tasks and Examples
A NALOGOUS TASK
Let’s use a word problem and particular numbers to analyze
the structure of the square of a binomial expression.
Chuck Norris has a vegetable garden that measures 4
feet long and 4 feet wide. One day, Steven Seagal
makes fun of how small Chuck’s garden is. After
Chuck stares at Steven until he explodes, Chuck adds
three feet to both the length and the width. Draw a
picture of Chuck’s old garden, the new one, and find
the area of each.
Conclusion
Background
Evil Forces
Role of Analogous Tasks and Examples
A NALOGOUS TASK
Let’s use a word problem and introduce a variable to further
analyze the structure of the square of a binomial expression.
Chuck Norris has a vegetable garden that measures x
feet long and x feet wide. One day, Steven Seagal
makes fun of how small Chuck’s garden is. After
Chuck stares at Steven until he explodes, Chuck adds
three feet to both the length and the width. Draw a
picture of Chuck’s old garden, the new one, and find
an expression that represents the area of each.
Conclusion
Background
Evil Forces
Role of Analogous Tasks and Examples
A NALOGOUS TASK
Let’s use a word problem and introduce another variable to
further analyze the structure of the square of a binomial
expression.
Chuck Norris has a vegetable garden that measures x
feet long and x feet wide. One day, Steven Seagal
makes fun of how small Chuck’s garden is. After
Chuck stares at Steven until he explodes, Chuck adds
y feet to both the length and the width. Draw a picture
of Chuck’s old garden, the new one, and find an
expression that represents the area of each.
Conclusion
Background
Evil Forces
Role of Analogous Tasks and Examples
A NALOGOUS TASK
Let’s use a word problem and introduce another variable to
further analyze the structure of the square of a binomial
expression.
Chuck Norris has a vegetable garden that measures x
feet long and y feet wide. One day, Steven Seagal
makes fun of how small Chuck’s garden is. After
Chuck stares at Steven until he explodes, Chuck adds
a feet to the length and b feet to the width. Draw a
picture of Chuck’s old garden, the new one, and find
an expression that represents the area of each.
Conclusion
Background
Evil Forces
Role of Analogous Tasks and Examples
A N INVITATION
For next week:
I
Think about a lesson you are going to teach.
I
Anticipate one or two common student difficulties.
I
Design a sequence of analogous tasks to bridge student
understanding.
I
Send an outline of your revised lesson to
[email protected] by next week’s meeting.
I
We will look them over and give you feedback at the
following week’s meeting.
Conclusion
Background
Evil Forces
Role of Analogous Tasks and Examples
C ONCLUSION
I
Choose a replacement value of the variable to turn an
intricate algebraic expression into a more concrete
analogous task
Conclusion
Background
Evil Forces
Role of Analogous Tasks and Examples
C ONCLUSION
I
I
Choose a replacement value of the variable to turn an
intricate algebraic expression into a more concrete
analogous task
ATTEND TO PRECISION!
Conclusion
Background
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
C ONCLUSION
I
I
Choose a replacement value of the variable to turn an
intricate algebraic expression into a more concrete
analogous task
ATTEND TO PRECISION!
I
I
Students can learn to use precise vocabulary. It begins with
you modeling it in the classroom!
Robust misconceptions will require long-term and
consistent efforts to overcome. Don’t be discouraged!
Background
Evil Forces
Role of Analogous Tasks and Examples
T HANKS FOR COMING !
Dr. Roger Fischer
Mathematics Instructor
Gallatin College – Montana State University
201 Hamilton Hall
Bozeman, MT 59717
[email protected]
Conclusion