retooling the ged math classroom for college success

1

Retooling the GED Math Classroom for College

Success

Network 2009Tom Mechem, GED State Chief Examiner

October 22, 2009

2

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

3

There can be no “Post-Secondary”

without the “Secondary.”

4

You have to pass the GED tests and earn the diploma;

but

You must also earn a post-secondary degree or official credential.

5

For both “Secondary” (GED) and “Post-Secondary”:

Math Is The Major Impediment

6

Tom Mechem’s “Hit List”(Items That Resonate)

•Notation•Order of Operations•Triangles & Angles•Ratio and Proportion•“Reverse” Algebra•Charts, Graphs, and Tables

7

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

8

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

9

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

10

“Algebra is the Gatekeeper”

11

You have to pass the GED Math Test;but

You also have to pass the ACCUPLACER Algebra Test

12

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

13

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.

14

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

15

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

16

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?

17

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

18

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.”

19

20

So What Are We Gonna Do?

21

First, What Are We NOT Going To Do?

22

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!!

23

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!!

24

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)

25

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

26

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]

27

A “Problem” Problem(basic arithmetic)

3 + 4 + 5 = + 2

28

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!!

29

We Could Create a Successful GED Math Program

(We Already Have!)

But No One Can Pass the ACCUPLACER

30

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)

31

“Algebra Skills Are the Single Most Accurate Predictor of College

Success.”

(Even if you’re an English major!)

32

“Incorporate GED Subject Matter into an Algebra

Course Designed to Develop Algebraic Habits of Mind.”

33

Tom Mechem’s “Hit List”(Items That Resonate)

•Notation•Order of Operations•Triangles & Angles•Ratio and Proportion•“Reverse” Algebra•Charts, Graphs, and Tables

34

Two Key Components

• Sound Pedagogical Principles (I know we have these)

• A Coherent and Integrated Curriculum(We have a ways to go)

35

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.

36

A Coherent Curriculum

• Depth, Rather Than Breadth

• Move Forward, Spiral Back

• Open, Rather Than Closed

37

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.

38

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?

39

Number Puzzle

X

+

40

Number Puzzle

3 2

X

+

41

Number Puzzle

24

6

X

+

42

Number Puzzle

8 1.5

X

+

43

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)

44

Fractions

2 = 4 = 18 16 4

(2)(1) = (2)(2) = 1(2)(2)(2) (2)(2)(2)(2) (2)

(2)

45

Signed Numbers

-3 + (-4)

46

GED Multi-Step Problems

Juan works in a cell phone store. He makes $200 a week plus $25 for every

cell phone he sells.

47

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.

48

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.

49

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.

50

Input Function Rule Output

Cell Phones YouSell in One Week

( )x25 + 200 Your

Weekly Pay

39

500625

Juan works in a cell phone store. He makes $200 a week plus $25 for every cell phone

he sells.

51

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

52

Distributive Property

4 (3 + 2)(could be considered Order of

Operations):4(5)but

4(3) + 4(2)prepares for

x(x + 3)

53

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.”

54

And Then Moving On To:

• Factoring

• Polynomials

• Quadratic equations

55

Do We Need a Different Class Design?

• More Intensity

• Greater Frequency

• Managed Enrollment

• College Prep Track

56

Tom Mechemtmechem@doe.mass.edu

781-338-6621