1 retooling the ged math classroom for college success network 2009 tom mechem, ged state chief...
TRANSCRIPT
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Retooling the GED Math Classroom for College
Success
Network 2009Tom Mechem, GED State Chief Examiner
October 22, 2009
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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
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There can be no “Post-Secondary”
without the “Secondary.”
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You have to pass the GED tests and earn the diploma;
but
You must also earn a post-secondary degree or official credential.
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For both “Secondary” (GED) and “Post-Secondary”:
Math Is The Major Impediment
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Tom Mechem’s “Hit List”(Items That Resonate)
•Notation•Order of Operations•Triangles & Angles•Ratio and Proportion•“Reverse” Algebra•Charts, Graphs, and Tables
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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
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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
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“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
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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.
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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
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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?
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8. If 2x – 3(x + 4) = - 5, then x = A. 7B. - 7C. 17D. – 17
9. – 3(5 – 6) – 4(2 – 3) =
A. - 7B. 7C. - 1D. 1
Sample ACCUPLACER Signed Number Problems
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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!!
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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!!
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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
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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]
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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!!
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We Could Create a Successful GED Math Program
(We Already Have!)
But No One Can Pass the ACCUPLACER
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We Could Put Together a Princeton-Review-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!)
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“Incorporate GED Subject Matter into an Algebra
Course Designed to Develop Algebraic Habits of Mind.”
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Tom Mechem’s “Hit List”(Items That Resonate)
•Notation•Order of Operations•Triangles & Angles•Ratio and Proportion•“Reverse” Algebra•Charts, Graphs, and Tables
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Two Key Components
• Sound Pedagogical Principles (I know we have these)
• A Coherent and Integrated Curriculum(We have a ways to go)
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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.
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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.
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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
+
40
Number Puzzle
3 2
X
+
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Number Puzzle
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6
X
+
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Number Puzzle
8 1.5
X
+
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GED Notation
Parentheses
• Grouping for Order of Operations: 4(3 +5)• To Indicate Multiplication: (4)(3)• To Separate a Number’s Sign from an
Operation8 + (-3)
• To indicate the Coordinates of an Ordered Pair
(4, -3)
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Fractions
2 = 4 = 18 16 4
(2)(1) = (2)(2) = 1(2)(2)(2) (2)(2)(2)(2) (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.
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Input Output
Cell Phones YouSell in One Week
Your Weekly
Pay2 250
Juan works in a cell phone store. He makes $200 a week plus $25 for every cell phone
he sells.
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Input Function Rule Output
Cell Phones YouSell in One Week
( )x25 + 200 Your
Weekly Pay
2 2505 3258 400
Juan works in a cell phone store. He makes $200 a week plus $25 for every cell phone
he sells.
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Input Function Rule Output
Cell Phones YouSell in One Week
( )x25 + 200 Your
Weekly Pay
01
Juan works in a cell phone store. He makes $200 a week plus $25 for every cell phone
he sells.
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Input Function Rule Output
Cell Phones YouSell in One Week
( )x25 + 200 Your
Weekly Pay
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500625
Juan works in a cell phone store. He makes $200 a week plus $25 for every cell phone
he sells.
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Commutative, Associative, & Distributive Properties
Commutativea+b = b+a
ab = ba
Associativea+(b+c) = (a+b)+c
a(bc)=(ab)c
Distributivea(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
