AMATYC 2011 Presentation Mathematical Literacy for College Students: Bringing New Life to Life Kathleen Almy, Heather Foes

2. The Problem

3. What is the purpose of
developmental mathematics?

4. From AMATYC’s
Beyond Crossroads
The curriculum of developmental mathematics programs should:
0 develop mathematical knowledge and skills so students can
successfully pursue their career goals, consider other career
goals, and function as successful citizens.
0 develop students’ study skills and workplace skills to enable
them to be successful in other courses and in their careers.
0 help students progress through their chosen curriculum as
quickly as possible.

5. RVC Developmental Math Redesign
The good:
Consistent, strong pass rates
Students best prepared for
college algebra; significant
improvement long-term
Not lab-based
The not so good:
Same content
Poor retention and application
No options for non-STEM
No improvement in non-STEM
class performance

6. The Problem: One Size Fits All
0 Does Intermediate Algebra make sense for all students?
0 Is developmental math just a checklist of skills to master?
0 Is the Calculus path right for everyone?
0 Could we do something different but just as rigorous?

7. Looking Forward, Looking Back
0 Developmental math is not about looking back and
recreating high school.
0 Developmental math is about looking forward to
prepare for college level coursework that the student
will take.
0 We shouldn’t let history guide all future decisions.

8. A Solution: MLCS

9. A New Course, A Simple Goal
In one semester, Mathematical Literacy for College Students
gives a student at the beginning algebra level the
mathematical maturity to be successful in statistics, liberal
arts math, or intermediate algebra.

10. Timeline for Change
2009 AMATYC New Life initiative began.
2010 New Life led to the Statway and Quantway (Mathway),
funded by Carnegie.
2011 19 Carnegie grant schools currently piloting Statway.
2011 Rock Valley College currently piloting MLCS.
2012 8 Carnegie grant schools will pilot MLCS.

11. Carnegie AMATYC
MLCS
Quantway New Life

12. Components of MLCS
0 Critical thinking
0 Reading & writing
0 Connections
0 Retention & understanding
0 21st century skills
0 Student success
0 Realistic situations
0 Rigor and high standards
Goal: Students will have the mathematical maturity and study skills to be
successful in their first college-level math class.

13. Using lessons learned through redesign
In-course advising
Cut scores
Intentional
design, assessment, c Variety of methods
ontinual
improvement
MML, student groups,
instructor help

14. Using research and experience
0 Researched schools, programs, and countries who are
effectively teaching mathematics (not just algebra)
0 Read and incorporated information on how the brain learns
0 Incorporated lessons learned in our redesign
0 Training sessions and materials
0 Advising
0 Materials
0 Online resources
0 Continual Assessment

15. A New Approach

16. It is the story that matters not just
the ending.
- Paul Lockhart

17. A New Perspective
0 Using the MLCS objectives, we wanted to build a developmental
course as we imagined it could be.
0 What would that look like?
0 Students doing and experiencing mathematics.
0 Skills are present but as a means to a greater end.
0 Situations are compelling, interesting, and real.
0 If this student goes to a statistics or general education math
class, what do we need them to know?
0 Throw out the old conventions and take a new perspective.
Mathematics is not a checklist; it’s an adventure.

18. Traditional Approach
0 Theory, then
applications if time
0 Each strand done
Proportions
Functions
separately to
Numbers
Algebra
completion
0 Algebra is primary
focus
0 Skill based
0 Examples of every
possible variation of
skill
(problem recognition)

19. New Approach
0 Applications to
motivate, then theory
as needed
Proportions
Functions
Numbers
Algebra
0 Strands addressed
each unit in an
integrated fashion
going deeper each
time
0 Equal time on each
strand
0 Concepts-based
Undercurrent of geometry, statistics,
0 Fewer skills, more student success, mathematical success
connections

20. Rules of Four: Approaches
0 Content
0 Numeracy, proportional
reasoning, algebraic
reasoning, functions
0 Problem solving (Polya)
0 Understand-Plan-Do-Look Back
0 Open-ended problem per unit
0 Each lesson
0 Representations
0 Verbal, numeric, algebraic, graphic

21. Content

22. The intuitive mind is a sacred gift and the
rational mind is a faithful servant. We have
created a society that honors the servant
and has forgotten the gift.
- Albert Einstein

23. Numeracy, then Algebra
0 The premise of using algebra to
illuminate how numbers work
doesn’t work. It obscures the THEME
point.
Emphasize
units. Numbers
0 Start with numbers and stay there are quantities.
for a while. Then generalize when
it makes sense to.
0 Stay concrete; stay tangible.

24. Algebraic Reasoning
0 Avoid naked problems (problems
without context) whenever
possible.
THEME
0 Use numeric methods until Judging when
students want and value the algebra makes
sense and how
algebraic method. to use it
0 Strive for meaningful situations and
variables.

25. Functional Focus
0 Functional relationships occur in
every unit.
0 We work on numeracy, algebraic THEME
reasoning, and proportions all the
while developing function Moving between
understanding. tables, graphs, a
0 Constant vs. variable nd equations
fluidly
0 Independent vs. dependent variable
0 Input values that make sense
0 We let students see that many
functions are not linear.

26. Proportional Reasoning
0 Proportional reasoning is much more than
“If 1 inch = 5 miles on a map, what does
7.5 inches equal?”
and THEME
“Cross multiply and divide.” Writing rates
with units and
scaling them
0 It’s a world of fractions, rates, making
sense of them, and seeing them in multiple
places in many ways.
0 Ratios and proportions have occurred in
nearly every lesson.

27. Choices, Choices
Less emphasis on… More emphasis on…
0 Factoring techniques 0 Numeracy
0 Worst-case scenario 0 Modeling
expressions and 0 Non-linear
equations functions, particularly
0 Naked problems exponential
0 Repetition 0 Messy, poorly framed
problems

28. What about factoring?
0 Could not cover all traditional algebra
content and do real-life problems in any
depth
0 Let some traditional topics go in favor of
more meaningful skills
0 Specific examples
0 Factoring GCF only
0 Build a quadratic function model
0 Build a rational function model
0 Develop statistical base from which to build
More advanced topics are addressed at an exposure level from a functions and
numerical perspective. Students can take intermediate algebra if more depth is
needed later.

29. Intentional Development
0 Slow and steady
0 By the time a topic is formalized, students have nearly
mastered it.
0 See a topic in multiple ways, multiple times, in multiple
contexts
0 Skills, concepts, applications in equal proportion
Example: Slope-intercept form was not introduced until after students had
numerous experiences generalizing a relationship from a table of data.

30. A New Perspective
0 We should not act as though these students have
never seen algebra because most have for years.
0 Instead, we approach content in new ways with a new
focus:
0 How does it work?
0 How can I use this?
0 When does this technique make sense?
Example: Most of our students could simplify an expression, but could not
write the expression from a situation.

31. Is this approach valuable?
0 Adjustment for everyone involved but the payoff is real
0 Doing real mathematics, not just skills
0 Open-ended problems, tough questions, Excel
0 Gone is the question, “When am I ever going to use this?”
0 Doing college-level work at a slower pace. Not high school
all over again.
0 Students seem to enjoy and appreciate the realism.

32. STEM vs. non-STEM
0 Course was built for the non-STEM student
0 Valuable to all students, especially STEM-bound ones
0 Developing scientific literacy
0 View topics through a math lens
0 “If you know the rules, you can play the game.”
0 Examples from chemistry, biology, physics

33. Seeing Mathematics

34. Students’ Preconceived Notions
0 “I already know all this.”
0 “I shouldn’t be expected to do
it unless you’ve shown me 10
examples like it.”
0 “You should be spelling
everything out more.”
0 “I shouldn’t have to work
more than an hour outside of
class each week.”

35. Teachers’ Preconceived Notions
0 Class is too easy and
will have high pass
rates.
0 Students aren’t learning
enough algebra and
won’t be ready for a
college-level course.
0 If you’re not doing all
the algebra, you’ve
lowered standards.
0 These students aren’t
capable of doing real
problems.

36. Approaching Content
0 Which skills will students need?
0 Where will they need to apply them?
0 How are these skills connected?

37. Integers and order of operations
Before After
Understand
Simplify: variation, build the
standard deviation
formula, use it to find
-3 – 2(-6 – 8) s.d. for a data
set, interpret it in
context
( x mean)2
s
n 1

38. Evaluating Expressions
Before After
Evaluate: Program cells in
Excel to do a task
3x – 2 when x = - 4

39. Linear Equations
Before After
Find slope, Build a cost model for
y-intercept, and a Kindle and Nook to
graph: compare against the
cost of a hardcover
book. When is each
y = -5x + 6 worth it? Use
graphs, equations, an
d tables.
N=179 + 12.99B
K=79 + 12.99B
H=35B

40. Plotting Points to Graph
Before After
Build a model. Plot points by
Make a t-table and
hand or Excel. Determine
graph a line:
shape and analyze.
Hours to pay for gallon of gas
7
6
5
4
3
2
1
0
0 10 20 30 40

41. Geometry
Before After
Find the volume of a If we overfill a medical
right circular cylinder measuring cup/spoon
whose height is 4 cm by 1 mm, which would
and diameter is 2 cm. produce a greater
overdose error?
Estimate volume in cc’s
and find actual and
percentage change.

42. The Role of Skills
A skill is not introduced until students see a need for it.
Online homework provides skill practice in a traditional
way, with and without context.
Skill questions without context still appear on
assessments to ensure students can perform them.
We spend less time on skills to have more time for
applications.

43. Attractive Mathematics
Useable
Examples: Biology, business, chemistry
Relevant
Example: Statistics through job figures
Interesting
Examples: Outwit & Outlast, viral videos
Outcome: MOTIVATION

44. Instruction

45. Don’t tell me the moon is shining; show
me the glint of light on broken glass.
- Anton Chekhov

46. Show, Don’t Tell
Lesson and Unit Protocol:
0 MOTIVATE: Explore an
interesting situation or hook
0 DEVELOP: Learn more about it
through activities, mini-lecture
(theory) , hands-on activities, etc.
0 CONNECT: Associate concepts
back and forward
0 REFLECT: Wrap-up topic Self-similarity
0 PRACTICE: online for
skills, paper for concepts &
applications

47. Addressing Quantway Goals
Engagement 0 Students actively work on
rich problems, both closed
and open-ended.
Connections 0 Students make sense of
topics in the given setting and
others.
Productive persistence 0 Students are allowed to
struggle, but assistance is
provided when necessary.
Deliberate practice 0 Students complete
homework assignments
which forge connections and
deepen conceptual
understanding.

48. Technology for the 21st Century
0 Mental arithmetic is encouraged
whenever possible.
0 Calculators are used when they are
needed.
0 Excel is used for analyzing patterns and
making graphs.

49. Striking the right balance
Need Engagement Frustration
enough
structure to
give
students Contextual Theoretical
comfort but
not so much Paper HW/By
that it is Online HW/Tech
hand
monotonous
Group Work Lecture
Open-ended Single solution

50. A Look Into MLCS

51. The Class
Heather working with a group of
students.
Students working together.

52. The Class
We still have whole class time
and some direct instruction.

53. What does a lesson
look like?
Snapshots of a strand

54. Topic: Solving equations
Lead up:
Built expressions and
equations
Graphed them
Solved numerically, with
Excel, or proportionally
Need more powerful
techniques as situations
Packet grow more complex

55. Goal:
Understand the
logic of the
equation
solving process

56. Goal:
Build and
solve 1-step
equations
Online
homework

57. Statistically
Goal: 3.10 On the rise using Pareto
charts, mean, med
Connect 1-step Read an article ian, standard
equations to other about food price deviation
situations and skills inflation/package
reduction and Algebraically
analyze it 3 ways. by building and
solving equations
to find original
prices and sizes
Geometrically
by analyzing changes
in
Online homework &
paper homework dimensions, volume,
and surface area.

58. 3.11 Hands-On Equations
Goal:
Solve 2-step and
multi-step linear
equations
Build approach
to equation
solving
Online
homework

59. Goal:
Build equations to 3.12
solve in an applied
setting
Connect equation
solving to previous Quarter
skills
Wing
Night

60. Numerically
Goal: 3.13 Eastbound using table of
and Down values
Visualize equations
and their qualities on a More expensive gas or Algebraically
graph cheaper gas with a car by building
wash? models and
determining when
Analyze two gas price the price would
options 3 ways. be the same
Graphically
by graphing the
functions and
Online homework & interpreting the
paper homework
solution to the
equation visually

61. The whole is greater than the sum of its parts.
-John Heywood

62. What does it feel like?
Participate in Quarter Wing Night lesson.
Packet

63. Outcomes
Unexpected twists and turns
Unusual combinations of content
Fun

64. A Look Into MLCS

65. Pilot: Our students
0 The grades are not as high as a typical beginning algebra
course.
0 Not about skills; it’s about problem solving.
0 Students swing from overconfident to overwhelmed in a
heartbeat.
0 Structured lessons in ways that reduce this.
0 They’re used to mimicking. We’re asking them to make sense
of mathematics.
0 They have to be taught how to study and succeed in this type of
course, which is like a college level class.
0 Students resisted at first, but are cooperative now.

66. Lessons from the Pilot:
A Charade
0 Traditional courses allow us to
maintain a distance.
0 When you probe beyond that, it
is disturbing how little they
really know.
0 Students learn to play the Mastery learning on online systems
game, but they’re not means little.
necessarily learning Prerequisite quizzing example
Application of skills issues
mathematics.

67. Lessons from the Pilot:
A Depressing Reality
0 Most of our students have taken 4 – 6 years of
algebra and yet placed into Beginning Algebra.
0 This course shows them what they do and do not
know.
0 We cannot help them all.
0 Low cognitive abilities
0 Some students need 1 year in developmental math
(but not all).

68. Lessons from the Pilot
0 A frame of reference and context go a long way in improving
connections and understanding.
0 Reflection is necessary to make sense of a lesson in the larger
scheme.
0 Letting things develop organically instead of prescriptively is
more engaging to students.
0 Students need accessible challenges to maintain interest.
0 Numbers are hard but helpful; generalizing is difficult but
necessary.
0 We are essentially “flipping” the classroom, which is refreshing.

69. Implementation

70. Considering redesign?
Consider MLCS

71. Implementation Ideas
Replace Beginning Algebra
STEM
Intermediate College
Algebra Level Math
Prealgebra MLCS
Non-STEM
College
Level Math
(Statistics, Libe
ral Arts Math)
Packet

72. Implementation Ideas
Use MLCS lessons in an emporium for once-weekly
problem solving sessions
Beginning Intermediate College
Prealgebra
Algebra Algebra Level Math
0 Previews content for some, connects for others
0 Everyone engaged
0 More than just skills

73. Implementation Ideas
Augment traditional sequence with MLCS
as a non-STEM alternative preparation
for statistics/liberal arts math.
STEM
Beginning Intermediate College
Algebra Algebra Level Math
Prealgebra
Non-STEM
MLCS College
Level Math
(Statistics, Libe
ral Arts Math)
Students who change their major can take
intermediate algebra as a bridge to STEM courses.

74. RVC Flowchart

75. How big, how much?
0 Course is 3 – 6 credit hours depending on your state
and school requirements.
0 Some topics (systems of linear equations, quadratic
modeling, rational modeling) are optional.
0 Great flexibility in terms of lessons and coverage.

76. Making MLCS Happen
0 Writing materials
0 Living textbook approach (sample in handouts)
0 Online & paper homework
0 Instructor notes throughout based on pilot so that anyone can
teach it
0 Team teaching (collaboratory)
0 Consider this if trying a pilot
0 Attractive, simpler option in a redesign
0 Addressing articulation

77. Q&A

78. For more information
Kathleen Almy
kathleenalmy@gmail.com
http://almydoesmath.blogspot.com
Heather Foes
Heather.foes@gmail.com
Packet