Transcript Preliminary Findings from the School-to

Retooling the GED Math
Classroom for College Success
Network 2009
Tom Mechem, GED State Chief Examiner
October 22, 2009
1
The Handwriting Is On The Wall
• $27.53/hour
• 9 out of 10 of the fastest-growing jobs in
MA require Associate’s Degree or higher
• “Two years of post-secondary education
are needed…to emerge out of poverty.” –
John Comings
2
There can be no
“Post-Secondary”
without the
“Secondary.”
3
You have to pass the GED tests and
earn the diploma;
but
You must also earn a post-secondary
degree or official credential.
4
For both “Secondary” (GED)
and “Post-Secondary”:
Math Is The Major
Impediment
5
Tom Mechem’s “Hit List”
(Items That Resonate)
• Notation
• Order of Operations
• Triangles & Angles
• Ratio and Proportion
• “Reverse” Algebra
• Charts, Graphs, and Tables
6
College
• 66% of GED Examinees say they are taking the
tests to get into college
• 27% of GED grads ever enroll in college
• Of those:
– Less than 1 in 5 completes two years of
college
– 4% ever complete four years of college
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One Culprit: Developmental Courses
• 85% of GED grads entering CC require at least
one developmental course
• If a CC entrant requires two or more
developmental courses, the chances are almost
nil that this person will earn an Associate’s
degree
8
Unindicted Co-Conspirator:
The ACCUPLACER Test
GED entrants vs. total incoming CC cohort
• Reading: GED entrants do better
• Writing: GED entrants do as well in avoiding
developmental courses; not as well in being
placed in advanced writing classes
• Math: GED entrants do much, much worse
9
“Algebra is the Gatekeeper”
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You have to pass the GED Math Test;
but
You also have to pass the ACCUPLACER
Algebra Test
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Survival on the ACCUPLACER
Students must be able to:
•
•
•
•
•
•
Add radicals and algebraic fractions
Evaluate algebraic expressions
Factor polynomials
Factor the difference of squares
Square binomials
Solve linear equations
12
Philosophies: GED vs. ACCUPLACER
• GED: “GED examinees represent a very diverse
population with respect to age, educational
background, and future goals. The item
contexts should reflect that.”
• ACCUPLACER: determine ability level for
college placement.
13
Two Math Problems
GED
“Big Papi Ortiz has a beautiful, grassy,
rectangular back yard that measures 120 feet
by 90 feet. He intends to build a square stone
patio in his yard. The patio will measure 60
feet on a side. Once the patio is built, how
many square feet of grass will he have in his
yard?”
7200 square feet
14
Two Math Problems (cont.)
ACCUPLACER
“A rectangle has a length of 2a + 4 and a width of
a – 3. If the formula for the area of a rectangle
is area = length x width, what is the area of
this rectangle?”
2a2 -2a -12
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Two Math Problems (cont.)
ACCUPLACER
1 + 1
y+3
y
=
The fraction x2 + 8x +16
x2 – 16
can be written as which of the following?
16
Sample ACCUPLACER Signed Number
Problems
8. If 2x – 3(x + 4) = - 5, then x =
A. 7
B. - 7
C. 17
D. – 17
9. – 3(5 – 6) – 4(2 – 3) =
A.
B.
C.
D.
-7
7
-1
1
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Philosophies: GED vs. ACCUPLACER
• GED: “If it doesn’t resonate, you can get
it wrong and survive.”
• ACCUPLACER: “If you get it wrong, you
are doomed.”
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So What Are We Gonna Do?
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First, What Are We NOT Going To Do?
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Common Knowledge,
Called Into Question,
v.1
“The higher you score on the GED tests, the
better chance you have of passing the
ACCUPLACER.”
Not True!!
22
Common Knowledge,
Called Into Question,
v.2
“Get your arithmetic skills up to par; then you
can move to algebra and have a better chance
of mastering algebraic skills.”
Not True!!
23
A Problem
Arithmetic skills are not a reliable predictor of
nor a necessary prerequisite for Algebra skills
(In fact, they may get in the way)
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Arithmetic May Be the Problem With
Algebra
This runs counter to intuition
• if you’re good at one aspect of math, you’ll be
good at another
And to traditional teaching practice
• Master arithmetic before being introduced to
Algebra
25
Two Reasons Arithmetic Is a Problem
• Limited concept of the “equals” sign
(“total” rather than “mathematical equivalence”)
(a “do something” sign rather than a relational
symbol)
• Limited concept of the “minus” sign
(only “subtract” rather than also “negative
something” [-3] and “the opposite of” [-x]
26
A “Problem” Problem
(basic arithmetic)
3+4+5=
+2
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Common Knowledge,
Called Into Question,
v.3
“If you fail the ACCUPLACER Algebra test, go into
an arithmetic remedial course. Success there
will prepare you to re-take and pass the
Algebra test.”
Not True!!
28
We Could Create a Successful GED
Math Program
(We Already Have!)
But No One Can Pass the ACCUPLACER
29
We Could Put Together a PrincetonReview-Type ACCUPLACER Prep Class
But No One Would Pass the GED Tests
(And They Wouldn’t Have the Habits of
Mind to Succeed in College Math)
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“Algebra Skills Are the Single Most
Accurate Predictor of College
Success.”
(Even if you’re an English major!)
31
“Incorporate GED Subject
Matter into an Algebra Course
Designed to Develop Algebraic
Habits of Mind.”
32
Tom Mechem’s “Hit List”
(Items That Resonate)
• Notation
• Order of Operations
• Triangles & Angles
• Ratio and Proportion
• “Reverse” Algebra
• Charts, Graphs, and Tables
33
Two Key Components
• Sound Pedagogical Principles
(I know we have these)
• A Coherent and Integrated Curriculum
(We have a ways to go)
34
Pedagogical Principles
(plagiarized from Steve Hinds)
• Math learning that is meaningful and not “rote-ful”
• Lecture is almost non-existent
• Rules can be the endpoint, not the starting point
• Use what they already know: Functions in context and
number relationships can illuminate more abstract
algebra ideas
• Ask students to think like scientists
• Student talk is more important than teacher talk –
questioning, alternate solutions, collaboration, student
errors.
35
A Coherent Curriculum
• Depth, Rather Than Breadth
• Move Forward, Spiral Back
• Open, Rather Than Closed
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A “Closed” Problem
Some money is shared between Maria and Ted so
that Maria gets $5 more than Ted gets. Ted gets
“x” dollars. Use algebra to write Maria’s
amount. The money to be shared is $47. Use
algebra to work out how much Maria and Ted
would each get.
37
Perhaps a Better Problem
(hard to do on your fingers and toes)
(open)
BANQUET TABLES
Arrangement 1
Arrangement 2
Arrangement 3
Arrangement 1 seats four people. How many people can
be seated at Arrangement 100?
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Number Puzzle
X
+
39
Number Puzzle
X
3
2
+
40
Number Puzzle
X
24
6
+
41
Number Puzzle
X
8
1.5
+
42
GED Notation
Parentheses
• Grouping for Order of Operations: 4(3 +5)
• To Indicate Multiplication: (4)(3)
• To Separate a Number’s Sign from an Operation
8 + (-3)
• To indicate the Coordinates of an Ordered Pair
(4, -3)
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Fractions
2=4=1
8 16 4
(2)(1)
(2)(2)(2)
= (2)(2) =
(2)(2)(2)(2)
1
(2)(2)
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Signed Numbers
-3 + (-4)
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GED Multi-Step Problems
Juan works in a cell phone
store. He makes $200 a week
plus $25 for every cell phone
he sells.
46
Juan works in a cell phone store. He makes $200
a week plus $25 for every cell phone he sells.
Input
Cell Phones You
Sell in One Week
2
Output
Your
Weekly
Pay
250
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Juan works in a cell phone store. He makes $200
a week plus $25 for every cell phone he sells.
Input
Function Rule
Cell Phones You
Sell in One Week
( )x25 + 200
2
5
8
Output
Your
Weekly
Pay
250
325
400
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Juan works in a cell phone store. He makes $200
a week plus $25 for every cell phone he sells.
Input
Function Rule
Cell Phones You
Sell in One Week
( )x25 + 200
Output
Your
Weekly
Pay
0
1
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Juan works in a cell phone store. He makes $200
a week plus $25 for every cell phone he sells.
Input
Function Rule
Cell Phones You
Sell in One Week
( )x25 + 200
Output
Your
Weekly
Pay
3
9
500
625
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Commutative, Associative, & Distributive
Properties
Commutative
a+b = b+a
ab = ba
Associative
a+(b+c) = (a+b)+c
a(bc)=(ab)c
Distributive
a(b+c)=ab+ac
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Distributive Property
4 (3 + 2)
(could be considered Order of Operations):
4(5)
but
4(3) + 4(2)
prepares for
x(x + 3)
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Graphing Linear Equations
x = 3y
but also:
“Juan works in a cell phone store.
He makes $200 a week plus $25
for every cell phone he sells.”
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And Then Moving On To:
• Factoring
• Polynomials
• Quadratic equations
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Do We Need a Different Class
Design?
• More Intensity
• Greater Frequency
• Managed Enrollment
• College Prep Track
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Tom Mechem
[email protected]
781-338-6621
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