From Memorizing to Understanding: Changing Developmental Mathematics Students’ Beliefs about Learning Wade Ellis, Jr. West Valley College

James Stigler: UCLA Psychology Dept. in May 2011 Math AMATYC Educator Students who have failed...[might succeed] if we can first convince them that mathematics makes sense... Students who have failed...[might succeed] if we can first convince them that mathematics makes sense the ability to correctly remember and execute procedures... is a kind of knowledge that is fragile without deeper conceptual understanding of fundamental mathematical ideas.... the ability to correctly remember and execute procedures... is a kind of knowledge that is fragile without deeper conceptual understanding of fundamental mathematical ideas. Finally, when students are able to provide conceptual understanding, they also produce correct answers. Finally, when students are able to provide conceptual understanding, they also produce correct answers.

What We Just Learned at Lunch from Jim Stigler All nations with highly successful math students have teachers that create environments where students can experience: All nations with highly successful math students have teachers that create environments where students can experience: Productive Struggle Productive Struggle Explicit Connections Explicit Connections Deliberate Practice Deliberate Practice

Outline What we know about improving performance What we know about improving performance Simple ideas about promoting understanding Simple ideas about promoting understanding More involved ideas More involved ideas An holistic approach to understanding An holistic approach to understanding Instructional Design Instructional Design Bloom’s Taxonomy (updated) Bloom’s Taxonomy (updated) Learning Process Methodology Learning Process Methodology An example of a lesson An example of a lesson Comments and Questions Comments and Questions

What We Know About Improving Performance

Basic Skills Initiative (BSI) The Big Five/Social aspects of a course The Big Five/Social aspects of a course Come to class, Come to class, On time, On time, Pay attention Pay attention Take notes, and Take notes, and Do your homework Do your homework Context-based activities Context-based activities High expectations High expectations Frequent feedback (MyMathLab) Frequent feedback (MyMathLab)

Adult Learners Malcolm Knowles makes the following assumptions about the design of learning for adult learners [Andragogy ] 1.Adults need to know why they need to learn something, 2.Adults need to learn experientially, 3.Adults approach learning as problem-solving, and 4.Adults learn best when the topic is of immediate value.

Neuroscience Research Act/Observe-Reflect/Conjecture/Test Act/Observe-Reflect/Conjecture/Test (David Kolb and James Zull) (David Kolb and James Zull) Emotions make a difference in retention Emotions make a difference in retention What students think of themselves What students think of themselves Social persuasion (Albert Bandura) Social persuasion (Albert Bandura)

The Dana Foundation Uri Treisman (specific to mathematics) Uri Treisman (specific to mathematics) Adaptive Reasoning Adaptive Reasoning Strategic Competence Strategic Competence Conceptual Understanding Conceptual Understanding Productive Disposition Productive Disposition Procedural Fluency Procedural Fluency

Simple Ideas

Dos and Don’ts Don’t say “That’s easy.” when asked to talk about a problem. Ask the students if they had asked a student. Don’t say “That’s easy.” when asked to talk about a problem. Ask the students if they had asked a 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 students. Do come to class early. Great for knowing 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 active. Don’t lecture the whole period. Let students be 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 Ideas Discuss how ideas are connected Discuss how ideas are connected Concepts Maps Concepts Maps

Variable Expressions Equations Functions Re-express Solve Graph Developmental Mathematics Analyze Interpret Verify

An Holistic Approach

An Approach that Applies What We Know Jim Stigler on Environment Jim Stigler on Environment BSI BSI Adult Learners Adult Learners Neuroscience Neuroscience Dana Foundation Dana Foundation

A Framework for Lesson Design Bloom’s Taxonomy of Learning Information - remembering Information - remembering Knowledge - understanding Knowledge - understanding Application - applying Application - applying Problem Solving - analyzing Problem Solving - analyzing Evaluation- judging Evaluation- judging Research - creating Research - creating

Learning Skills Examples of Learning Skills that can be improved Examples of Learning Skills that can be improved Validating results Validating results Using prior knowledge Using prior knowledge Observing Observing Constructing examples Constructing examples Abstracting Abstracting Generalizing Generalizing Persisting Persisting Accepting responsibility Accepting responsibility Learning as a Process that can be improved

Creating a Lesson for a “Stigler” Environment Seeing the textbook as a resource, but not the course Seeing the textbook as a resource, but not the course Infusing a lessons with new knowledge of learning, neuroscience, and culture Infusing a lessons with new knowledge of learning, neuroscience, and culture Deciding on the behaviors you want the students to display as a result of the course Deciding on the behaviors you want the students to display as a result of the course Backward design (Carol Twigg, NCAT, & Redesign) Backward design (Carol Twigg, NCAT, & Redesign) Concept Maps (made by students) Concept Maps (made by students)

Learning Process Methodology 1. Preparation 2. Learning Activity 3. Reflection

1. Preparation Why? Why? Where does it fit in the knowledge framework? Where does it fit in the knowledge framework? Prerequisites Prerequisites Performance Goals Performance Goals Performance Criteria Performance Criteria Language/Vocabulary Language/Vocabulary Information (Resources for Learning) Information (Resources for Learning)

2. Learning Activity Plan Plan Key Questions/Critical Thinking Questions (Inquiry Questions encouraging Reflection) Key Questions/Critical Thinking Questions (Inquiry Questions encouraging Reflection) Examples and Models Examples and Models Application Application Problem Solving Problem Solving

3. Reflection Self-Assessment Self-Assessment Extension Extension

A Lesson Simplifying Algebraic Expressions

1. Preparation Why? 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 operator Information Key Concepts Methodologies Addressing Common Errors Preparatory Inventory

2. Learning Activity Plan 1. Discuss the questions from the Preparatory Inventory 2. Read the Critical Thinking Questions in Teams 4. Answer the Critical Thinking Questions 3. Solve the Application Exercises 4. Solve the Problems Examples and Models Simplify: 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

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 Contour Map Contour Map Contour Map Contour Map

1. Preparation Why? 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 map Informaiton Contour map animation

2. Learning Activity Plan 1. Read the Critical Thinking Questions 2. Work with the Contour map program 4. Answer the Critical Thinking Questions 3. Interpret the Applications 4. Solve the Problems Examples and Models Contour 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?

Comments and Questions

Additional Information (choose Literature Review) Basic Skills as a Foundation for Student Success in California Community Colleges (choose Literature Review) Basic Skills Handbook (choose Basic Skills Handbook) James Zull, ( 2002). The Art of Changing the Brain: Enriching the Practice of Teaching by Exploring the Biology of Learning. James Zull, ( 2002). The Art of Changing the Brain: Enriching the Practice of Teaching by Exploring the Biology of Learning. Knowles, M. (1984). Andragogy in Action. San Francisco: Jossey-Bass. Knowles, M. (1984). Andragogy in Action. San Francisco: Jossey-Bass. Dan Apple, and others (2009). Electronic Faculty Guidebook. Lisle, Illinois: Pacific Crest. Dan Apple, and others (2009). Electronic Faculty Guidebook. Lisle, Illinois: Pacific Crest. Contour Map Contour Maphttp://curvebank.calstatela.edu/contour81/contour81.htm