Retooling the GED Math Classroom for College Success

1. Retooling the GED Math Classroom for College Success North Shore Community College Tom Mechem, GED State Chief Examiner June 10, 2010

2. The Handwriting Is On The Wall • $29.60/hour • Twice as many jobs over the next decade will require a postsecondary credential or a college degree – Bureau of Labor Statistics • “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. “Cognitive skills and educational credentials are the strongest predictors of success in the labor markets.”

5. You have to pass the GED tests and earn the diploma; but You must also earn a post-secondary degree or official credential.

6. For both “Secondary” (GED) and “Post-Secondary”: Math Is The Major Impediment

7. Secondary(GED) Of MA GED Non-Passers whose literacy level is high enough to pass the GED tests, 62% fail because they fail Math only.

8. Tom Mechem’s “Hit List”(Items That Resonate) • Notation • Order of Operations • Triangles & Angles • Ratio and Proportion • “Reverse” Algebra • Charts, Graphs, and Tables

9. Post-Secondary(based on a 2008 GEDTS study) • 66% of GED Examinees say they are taking the tests to get into college • 37% of GED grads ever enroll in college • Of those: • 77% last only one semester • 5.5% ever earn any kind of a post-secondary degree or certificate In other words, only 2% of GED grads ever earn a post-secondary degree or certificate.

10. 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

11. 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

12. ACCUPLACER Math Test Three Levels • College Math – take for college-course placement if you do well the Algebra test • Algebra – must achieve a certain score to avoid developmental courses • Arithmetic – cannot avoid developmental courses no matter how well you do

13. 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

14. 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.

15. 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

16. 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

17. 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?

18. 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. - 7 B. 7 C. - 1 D. 1

19. 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.”

21. So What Are We Gonna Do?

22. First, What Are We NOT Going To Do?

23. 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!!

24. 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!!

25. 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)

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. “Algebra is the Gatekeeper” • You must pass the Algebra Accuplacer test to avoid developmental courses • Algebra Skills Are the Single Most Accurate Predictor of College Success(Even if you’re an English major!) • “STEM skills are required for an individual to be a 21st-Century literate person…”

29. “Incorporate GED Subject Matter into an Algebra Course Designed to Develop Algebraic Habits of Mind.”

30. Algebraic Habits of Mind • Building Rules to Represent Functions • Backwards – Forwards (Doing and Undoing) • Abstracting from Computation

31. Academic (Life?) Habits of Mind“Self-Confidence is Born of Demonstrated Ability” • Persisting • “Cognitive Flexibility” • Thinking About Thinking (Metacognition) • Questioning • Striving for Accuracy/Precision • Having the Confidence to Make Mistakes

32. “How We Teach Is As Important as What We Teach.” “Curriculum Development Is Professional Development.”

33. Two Key Components • Sound Pedagogical Principles • A Coherent and Integrated Curriculum

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

36. 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?

38. Tom Mechem’s “Hit List”(Items That Resonate) Notation Order of Operations Triangles & Angles Ratio and Proportion “Reverse” Algebra Charts, Graphs, and Tables

39. 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)

40. X + Building Arithmetic Skills

41. X 3 2 + Building Arithmetic Skills

42. X 24 6 + Building Arithmetic Skills

43. X 8 1.5 + Building Arithmetic Skills

44. Fractions 2 = 4 = 1 8 16 4 (2)(1) = (2)(2) = 1 (2)(2)(2) (2)(2)(2)(2) (2)(2)

45. Signed Numbers -3 + (-4) -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. Juan works in a cell phone store. He makes $200 a week plus $25 for every cell phone he sells.

48. 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

49. Distributive Property 4 (3 + 2) (could be considered Order of Operations): 4(5) but 4(3) + 4(2) prepares for x(x + 3)

50. 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.”