What is the purpose ofdevelopmental mathematics?
From AMATYC’s Beyond CrossroadsThe 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.
RVC Developmental Math RedesignThe good:Consistent, strong pass ratesStudents best prepared forcollege algebra; significantimprovement long-termNot lab-basedThe not so good:Same contentPoor retention and applicationNo options for non-STEMNo improvement in non-STEMclass performance
The Problem: One Size Fits All0 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?
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.
A New Course, A Simple GoalIn one semester, Mathematical Literacy for College Studentsgives a student at the beginning algebra level themathematical maturity to be successful in statistics, liberalarts math, or intermediate algebra.
Timeline for Change2009 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.
Components of MLCS0 Critical thinking0 Reading & writing0 Connections0 Retention & understanding0 21st century skills0 Student success0 Realistic situations0 Rigor and high standardsGoal: Students will have the mathematical maturity and study skills to be successful in their first college-level math class.
Using lessons learned through redesign In-course advisingCut scoresIntentionaldesign, assessment, c Variety of methodsontinualimprovement MML, student groups, instructor help
Using research and experience0 Researched schools, programs, and countries who are effectively teaching mathematics (not just algebra)0 Read and incorporated information on how the brain learns0 Incorporated lessons learned in our redesign 0 Training sessions and materials 0 Advising 0 Materials 0 Online resources 0 Continual Assessment
It is the story that matters not justthe ending. - Paul Lockhart
A New Perspective0 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.
Traditional Approach0 Theory, then applications if time0 Each strand done Proportions Functions separately to Numbers Algebra completion0 Algebra is primary focus0 Skill based0 Examples of every possible variation of skill (problem recognition)
New Approach0 Applications to motivate, then theory as needed Proportions Functions Numbers Algebra0 Strands addressed each unit in an integrated fashion going deeper each time0 Equal time on each strand0 Concepts-based Undercurrent of geometry, statistics,0 Fewer skills, more student success, mathematical success connections
Rules of Four: Approaches0 Content 0 Numeracy, proportional reasoning, algebraic reasoning, functions0 Problem solving (Polya) 0 Understand-Plan-Do-Look Back 0 Open-ended problem per unit 0 Each lesson0 Representations 0 Verbal, numeric, algebraic, graphic
The intuitive mind is a sacred gift and therational mind is a faithful servant. We havecreated a society that honors the servantand has forgotten the gift. - Albert Einstein
Numeracy, then Algebra0 The premise of using algebra to illuminate how numbers work doesn’t work. It obscures the THEME point. Emphasize units. Numbers0 Start with numbers and stay there are quantities. for a while. Then generalize when it makes sense to.0 Stay concrete; stay tangible.
Algebraic Reasoning0 Avoid naked problems (problems without context) whenever possible. THEME0 Use numeric methods until Judging when students want and value the algebra makes sense and how algebraic method. to use it0 Strive for meaningful situations and variables.
Functional Focus0 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 sense0 We let students see that many functions are not linear.
Proportional Reasoning0 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 them0 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.
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 buildMore advanced topics are addressed at an exposure level from a functions andnumerical perspective. Students can take intermediate algebra if more depth isneeded later.
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 proportionExample: Slope-intercept form was not introduced until after students had numerous experiences generalizing a relationship from a table of data.
A New Perspective0 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.
Is this approach valuable?0 Adjustment for everyone involved but the payoff is real0 Doing real mathematics, not just skills 0 Open-ended problems, tough questions, Excel0 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.
STEM vs. non-STEM0 Course was built for the non-STEM student0 Valuable to all students, especially STEM-bound ones0 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
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.”
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.
Approaching Content0 Which skills will students need?0 Where will they need to apply them?0 How are these skills connected?
Integers and order of operations Before After UnderstandSimplify: 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
Evaluating Expressions Before AfterEvaluate: Program cells in Excel to do a task3x – 2 when x = - 4
Linear Equations Before AfterFind slope, Build a cost model fory-intercept, and a Kindle and Nook tograph: compare against the cost of a hardcover book. When is eachy = -5x + 6 worth it? Use graphs, equations, an d tables. N=179 + 12.99B K=79 + 12.99B H=35B
Plotting Points to Graph Before After Build a model. Plot points byMake a t-table and hand or Excel. Determinegraph 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
Geometry Before AfterFind the volume of a If we overfill a medicalright circular cylinder measuring cup/spoonwhose height is 4 cm by 1 mm, which wouldand diameter is 2 cm. produce a greater overdose error? Estimate volume in cc’s and find actual and percentage change.
The Role of SkillsA skill is not introduced until students see a need for it.Online homework provides skill practice in a traditionalway, with and without context.Skill questions without context still appear onassessments to ensure students can perform them.We spend less time on skills to have more time forapplications.
Don’t tell me the moon is shining; showme the glint of light on broken glass. - Anton Chekhov
Show, Don’t TellLesson and Unit Protocol:0 MOTIVATE: Explore an interesting situation or hook0 DEVELOP: Learn more about it through activities, mini-lecture (theory) , hands-on activities, etc.0 CONNECT: Associate concepts back and forward0 REFLECT: Wrap-up topic Self-similarity0 PRACTICE: online for skills, paper for concepts & applications
Addressing Quantway GoalsEngagement 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.
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.
Striking the right balanceNeed Engagement Frustrationenoughstructure togivestudents Contextual Theoreticalcomfort butnot so much Paper HW/Bythat it is Online HW/Tech handmonotonous Group Work Lecture Open-ended Single solution
StatisticallyGoal: 3.10 On the rise using Pareto charts, mean, medConnect 1-step Read an article ian, standardequations to other about food price deviationsituations 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.
Goal:Build equations to 3.12solve in an appliedsettingConnect equationsolving to previous Quarterskills Wing Night
NumericallyGoal: 3.13 Eastbound using table of and Down valuesVisualize equationsand their qualities on a More expensive gas or Algebraicallygraph 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
The whole is greater than the sum of its parts. -John Heywood
What does it feel like? Participate in Quarter Wing Night lesson.Packet
OutcomesUnexpected twists and turnsUnusual combinations of contentFun
Pilot: Our students0 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.
Lessons from the Pilot: A Charade0 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.
Lessons from the Pilot: A Depressing Reality0 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).
Lessons from the Pilot0 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.
Implementation IdeasReplace Beginning Algebra STEM Intermediate College Algebra Level Math Prealgebra MLCS Non-STEM College Level Math (Statistics, Libe ral Arts Math) Packet
Implementation IdeasUse MLCS lessons in an emporium for once-weeklyproblem 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
Implementation IdeasAugment traditional sequence with MLCSas a non-STEM alternative preparationfor 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.
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.
Making MLCS Happen0 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 it0 Team teaching (collaboratory) 0 Consider this if trying a pilot0 Attractive, simpler option in a redesign0 Addressing articulation