retooling the ged math classroom for college success

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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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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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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

+

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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


( )x25 + 200 Your

Weekly Pay

2 2505 3258 400


he sells.

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( )x25 + 200 Your

Weekly Pay

01


he sells.

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( )x25 + 200 Your

Weekly Pay

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500625


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

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Tom [email protected]

781-338-6621

retooling the ged math classroom for college success

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