from memorizing to understanding: changing developmental mathematics students’ beliefs about...
TRANSCRIPT
From Memorizing to Understanding:
Changing Developmental Mathematics
Students’ Beliefs about LearningWade Ellis, Jr.Wade Ellis, Jr.
West Valley CollegeWest Valley College
[email protected]@ti.com
James Stigler: James Stigler: UCLA Psychology UCLA Psychology Dept.Dept.
in May 2011 in May 2011 MathMathAMATYC AMATYC EducatorEducator Students who have failed . . .[might Students who have failed . . .[might
succeed] if we can first convince them succeed] if we can first convince them that mathematics makes sense . . . that mathematics makes sense . . .
. . . the ability to correctly remember . . . the ability to correctly remember and execute procedures . . . is a kind of and execute procedures . . . is a kind of knowledge that is knowledge that is fragilefragile without without deeper conceptual understanding of deeper conceptual understanding of fundamental mathematical ideas.fundamental mathematical ideas.
Finally, when students are able to Finally, when students are able to provide conceptual understanding, provide conceptual understanding, they also produce correct answers.they also produce correct answers.
What We Just Learned What We Just Learned at Lunch from Jim at Lunch from Jim
StiglerStigler All nations with highly successful All nations with highly successful
math students have teachers that math students have teachers that create environments where students create environments where students can experience:can experience: Productive StruggleProductive Struggle Explicit ConnectionsExplicit Connections Deliberate PracticeDeliberate Practice
OutlineOutline What we know about improving performanceWhat we know about improving performance Simple ideas about promoting understandingSimple ideas about promoting understanding More involved ideasMore involved ideas An holistic approach to understandingAn holistic approach to understanding
Instructional Design Instructional Design Bloom’s Taxonomy (updated)Bloom’s Taxonomy (updated) Learning Process MethodologyLearning Process Methodology An example of a lessonAn example of a lesson
Comments and QuestionsComments and Questions
What We Know About What We Know About Improving PerformanceImproving Performance
Basic Skills Initiative Basic Skills Initiative (BSI)(BSI)
The Big Five/Social aspects of a courseThe Big Five/Social aspects of a course Come to class,Come to class, On time, On time, Pay attentionPay attention Take notes, and Take notes, and Do your homework Do your homework
Context-based activitiesContext-based activities High expectationsHigh expectations Frequent feedback (MyMathLab)Frequent feedback (MyMathLab)
Adult LearnersAdult Learners
Malcolm Knowles makes the following Malcolm Knowles makes the following assumptions about the design of learning assumptions about the design of learning for adult learners for adult learners [Andragogy[Andragogy] ]
1.1. Adults need to know why they need to Adults need to know why they need to learn something, learn something,
2.2. Adults need to learn experientially, Adults need to learn experientially, 3.3. Adults approach learning as problem-Adults approach learning as problem-
solving, and solving, and 4.4. Adults learn best when the topic is of Adults learn best when the topic is of
immediate value.immediate value.
Neuroscience ResearchNeuroscience Research
Act/Observe-Reflect/Conjecture/TestAct/Observe-Reflect/Conjecture/Test (David Kolb and James Zull)(David Kolb and James Zull)
Emotions make a difference in retentionEmotions make a difference in retention What students think of themselvesWhat students think of themselves Social persuasion (Albert Bandura)Social persuasion (Albert Bandura)
The Dana FoundationThe Dana Foundation
Uri Treisman (specific to Uri Treisman (specific to mathematics)mathematics) Adaptive ReasoningAdaptive Reasoning Strategic CompetenceStrategic Competence Conceptual UnderstandingConceptual Understanding Productive DispositionProductive Disposition Procedural FluencyProcedural Fluency
Simple Ideas Simple Ideas
Dos and Don’tsDos and Don’ts Don’t say “That’s easy.” when asked to talk about Don’t say “That’s easy.” when asked to talk about
a problem. Ask the students if they had asked a a problem. Ask the students if they had asked a student.student.
Do return tests the next class meeting.Do return tests the next class meeting. Don’t use green or red colored chalk or markers.Don’t use green or red colored chalk or markers. Do come to class early. Great for knowing Do come to class early. Great for knowing
students.students. Don’t answer questions students can answer.Don’t answer questions students can answer. Do use lectures to wrap up learning.Do use lectures to wrap up learning. Don’t lecture the whole period. Let students be Don’t lecture the whole period. Let students be
active.active. Do acknowledge good thinking.Do acknowledge good thinking. Don’t give partial credit. Don’t give partial credit. Do train your tutors not to pick-up a pencil.Do train your tutors not to pick-up a pencil. Do ask students to explain or verify.Do ask students to explain or verify.
(Yes, you should, judiciously.)
More Involved IdeasMore Involved Ideas
Discuss how ideas are connectedDiscuss how ideas are connected Concepts MapsConcepts Maps
Variable
ExpressionsEquations
Functions
Re-express
Solve
Graph
Developmental Mathematics
Analyze
Interpret
Verify
An Holistic ApproachAn Holistic Approach
An Approach that AppliesAn Approach that Applies What We Know What We Know
Jim Stigler on EnvironmentJim Stigler on Environment BSIBSI Adult LearnersAdult Learners NeuroscienceNeuroscience Dana FoundationDana Foundation
A Framework for Lesson A Framework for Lesson Design Design
Bloom’s Taxonomy of Bloom’s Taxonomy of LearningLearning
Information Information - remembering- remembering Knowledge Knowledge - understanding- understanding Application Application - applying- applying Problem Solving Problem Solving - analyzing- analyzing EvaluationEvaluation - judging- judging Research Research - creating - creating
Learning SkillsLearning Skills Examples of Learning Skills that can be Examples of Learning Skills that can be
improvedimproved Validating resultsValidating results Using prior knowledgeUsing prior knowledge ObservingObserving Constructing examplesConstructing examples AbstractingAbstracting GeneralizingGeneralizing PersistingPersisting Accepting responsibilityAccepting responsibility
Learning as a Process that can be Learning as a Process that can be improvedimproved
Creating a Lesson for a Creating a Lesson for a “Stigler” Environment“Stigler” Environment
Seeing the textbook as a resource, but Seeing the textbook as a resource, but not the coursenot the course
Infusing a lessons with new knowledge of Infusing a lessons with new knowledge of learning, neuroscience, and culturelearning, neuroscience, and culture
Deciding on the behaviors you want the Deciding on the behaviors you want the students to display as a result of the students to display as a result of the coursecourse Backward design (Carol Twigg, NCAT, & Backward design (Carol Twigg, NCAT, &
Redesign)Redesign) Concept Maps (made by students)Concept Maps (made by students)
Learning Process Learning Process MethodologyMethodology
1.1. PreparationPreparation
2.2. Learning ActivityLearning Activity
3.3. ReflectionReflection
1. Preparation1. Preparation
Why?Why? Where does it fit in the knowledge Where does it fit in the knowledge
framework?framework? PrerequisitesPrerequisites Performance GoalsPerformance Goals Performance CriteriaPerformance Criteria Language/VocabularyLanguage/Vocabulary Information Information (Resources for Learning)(Resources for Learning)
2. Learning Activity2. Learning Activity
PlanPlan Key Questions/Critical Thinking Key Questions/Critical Thinking
Questions Questions (Inquiry Questions encouraging Reflection)(Inquiry Questions encouraging Reflection)
Examples and ModelsExamples and Models ApplicationApplication Problem SolvingProblem Solving
3. Reflection3. Reflection
Self-AssessmentSelf-Assessment ExtensionExtension
A LessonA LessonSimplifying Algebraic Simplifying Algebraic
ExpressionsExpressions
1. PreparationWhy?
Part of mastering the language of mathematics involves writing algebra using the fewest possible symbols.
Where does it fit in the knowledge framework?You have learned about types of numbers that variables can take on and how variables are used in expressions. Now you will learn how to manipulate expressions. You will use this skill in solving equations.
Prerequisites How des the Commutative Property differ from the Associative Property? When do you use the Distributive Property? Give an example of where the Order of Operations must be used.Goals Change an expression into an equivalent expression.Performance Criteria Simplify an expression by removing parentheses and combining like terms.Language/Vocabulary binary operator, like terms, unary operatorInformation Key Concepts Methodologies Addressing Common Errors Preparatory Inventory
2. Learning ActivityPlan
1. Discuss the questions from the Preparatory Inventory2. Read the Critical Thinking Questions in Teams4. Answer the Critical Thinking Questions3. Solve the Application Exercises4. Solve the Problems
Examples and ModelsSimplify:
Key Questions/Critical Thinking Questions (Inquiry Questions)
What are equivalent expressions? What changes can you make to an expression to yield an equivalent expression? Why do you remove parentheses? How do you ensure that all terms are used in a simplification?Application Demonstrate Your Understanding: Problem Solving Interpret a word problem to create an expression and simplify it Create and Solve the Hardest Problem
2 21. 5 3 ( 2) 2. 2( 4) 3 ( 4)
3. 3 2(5 ) (5 ) 7
a a a x x x
x x x x
3 (6 2 ) 3 3( 5)x x x x
3. Reflection Self-Assessment: What did you learn about learning math? What did you learn about working with algebraic
expressions? How did you use the methodologies and common
errors in working through the DYU Problems?
What method will you use to remind yourself of important concepts dealing with simplifying algebraic expressions?
What learning skills did you use in this activity?Extension What kinds of expressions have you seen in your life? How have you used expressions in your life?
The Contour Map The Contour Map
Contour Map
1. PreparationWhy?
Understanding contour mapping will help you in reading such maps when you deciding on paths for power lines or hiking paths.
Where does it fit in the knowledge framework?You can currently read road maps and hiking trail maps. The ability to understand topographic information superimposed on these maps can allow you to understand the placement of buildings or plan hikes more effectively.
Prerequisites The ability to read street and road maps. An understanding of the coordinate plane.Goals The ability to interpret topographic maps. The ability to create a simple topographic map from data.Performance Criteria Reading topographic maps.Language/Vocabulary Slicing plane Level curve Contour Contour mapInformaiton Contour map animation
2. Learning ActivityPlan
1. Read the Critical Thinking Questions2. Work with the Contour map program4. Answer the Critical Thinking Questions3. Interpret the Applications4. Solve the Problems
Examples and ModelsContour Maps (see below)
Key Questions/Critical Thinking Questions (Inquiry Questions)
Why are the slicing planes the same distance apart? What does it mean when the contour lines are close together on the 2D contour map?
How can you tell which peak is highest from the contour plot?
Could the contour plot be of a set of valleys rather than a set of peaks?
What does it mean when there is a large region at the top of a peak?
Application Another topographic map to interpret (see below)Problem Solving
Decide on the steepest ascent to the highest mountain or ridge. Interpret a temperature contour map.
3. Reflection Self-Assessment Can you determine the steepest part of a
topographical map? Can you determine the saddle points of a
topographical map? Can you determine where the lakes or plains are on a
topographical map?
Extension To what other measurements can contour maps be
applied? Why would such maps be useful?
Additional InformationAdditional Information Basic Skills as a Foundation for Student Success in
California Community Collegeshttp://cccbsi.org/publications (choose Literature Review)http://cccbsi.org/publications (choose Literature Review)
Basic Skills Handbookhttp://cccbsi.org/publications (choose Basic Skills http://cccbsi.org/publications (choose Basic Skills Handbook)Handbook)
James Zull, ( 2002). James Zull, ( 2002). The Art of Changing the Brain: The Art of Changing the Brain: Enriching the Practice of Teaching by Exploring Enriching the Practice of Teaching by Exploring the Biology of Learningthe Biology of Learning..
Knowles, M. (1984). Knowles, M. (1984). Andragogy in ActionAndragogy in Action. San . San Francisco: Jossey-Bass. Francisco: Jossey-Bass.
Dan Apple, and others (2009). Dan Apple, and others (2009). Electronic Faculty Electronic Faculty GuidebookGuidebook. Lisle, Illinois: Pacific Crest.. Lisle, Illinois: Pacific Crest. http://www.pcrest.com/PC/FGB/4th/index.htm
Contour MapContour Maphttp://curvebank.calstatela.edu/contour81/contour81.htmhttp://curvebank.calstatela.edu/contour81/contour81.htm