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Kien Lim
Department of Mathematical Sciences
University of Texas at El Paso
Helping Students Develop
Mathematical Process Skills, Really?
CAMT Conference June 24, 2015
Introduction
Does this sound familiar?
“It’s weird!
Almost all my students aced this type of problem
in the test.
Yet most of them got it wrong in the final exam.”
Why is this phenomenon common?
What’s happening?
Introduction
CAMT Conference June 24, 2015
What are in the Texas College and Career Readiness Standards?
What are mathematical process skills?
Are they for real, or for embellishment?
How can we help our students develop these process skills?
Is it realistic?
Is there an essence underlying all the process standards?
If yes, what is it?
What need to change?
Are you really game for a change?
What support do you need?
Mathematical Process Skills
CAMT Conference June 24, 2015
Mathematical Process Skills
CAMT Conference June 24, 2015
Mathematical Process Skills
CAMT Conference June 24, 2015
;
;
;
Mathematical Process Skills
CAMT Conference June 24, 2015
; ;
1. Numbers and Operations2. Algebra3. Geometry4. Measurement5. Data analysis and Probability
1. Problem solving2. Reasoning and Proof3. Communication4. Representation5. Connections
Mathematical Process Skills
CAMT Conference June 24, 2015
; ;
1. Numbers and Operations2. Algebra3. Geometry4. Measurement5. Data analysis and Probability
1. Problem solving2. Reasoning and Proof3. Communication4. Representation5. Connections
CONTENT STANDARDS PROCESS STANDARDS
Are Both Sets Necessary? Why?
Mathematical Process Skills
CAMT Conference June 24, 2015
CONTENT STANDARDS PROCESS STANDARDS“The first subset is a collection, or structure, of structures consisting of particular axioms, definitions, theorems, proofs, problems, and solutions.
This subset consists of all the institutionalized ways of understanding in mathematics throughout history.”
(Harel, 2008, p. 272)
“The second subset consists of all the ways of thinking, which characterize the mental acts whose products comprise the first set.”
(Harel, 2008, p. 273)
Are Both Sets Necessary? Why?
1. “Students develop ways of thinking through the production of
ways of understanding, and, conversely,
2. The ways of understanding they produce are impacted by the
ways of thinking they possess.” (Harel, 2008, p. 272)
Harel’s Duality Principle
Mathematical Process Skills
CAMT Conference June 24, 2015
CONTENT STANDARDS PROCESS STANDARDS
Interconnected Knowledge
of Mathematics
Mathematical
Habits of Mind
Mathematics as a Discipline
Mathematical Process Skills
CAMT Conference June 24, 2015
CONTENT STANDARDS PROCESS STANDARDS
Interconnected Knowledge
of Mathematics
Mathematical
Habits of Mind
Is Texas really trying to focus on
process standards?
Mathematical Process Skills
CAMT Conference June 24, 2015
Revised TEKS – High School Math PROCESS STANDARDS
1. Problem solving
2. Reasoning and Proof
3. Communication
4. Representation
5. Connections
Mathematical Process Skills
CAMT Conference June 24, 2015
Common Core State Standards for Mathematics
PROCESS STANDARDS
TCCRS
Mathematical Process Skills
CAMT Conference June 24, 2015
TEKS – High School Math PROCESS STANDARDS
TEKS specifies what a student is expected to be able to do.
A major
difference between
TEKS and CCSSM:
Mathematical Process Skills
CAMT Conference June 24, 2015
Common Core State Standards for Mathematics
PROCESS STANDARDS
A major
difference between
TEKS and CCSSM:
CCSSM describes what a mathematically proficient student can do.
TEKS specifies what a student is expected to be able to do.
Mathematical Process Skills
CAMT Conference June 24, 2015
Common Core State Standards for Mathematics
PROCESS STANDARDS
1. Problem solving2. Reasoning and Proof3. Communication4. Representation5. Connections
Mathematical Process Skills
CAMT Conference June 24, 2015
PROCESS STANDARDS
Which set is easier to teach? Why?
Mathematical
Habits of MindInterconnected Knowledge
of Mathematics
CONTENT STANDARDS
Helping Students Develop Process Skills
CAMT Conference June 24, 2015
1 minute
How can we help students develop
mathematical process skills?
2 minutes
Now!
A You-Tube Video Break
Helping Students Develop Process Skills
CAMT Conference June 24, 2015
Dan Meyer on Real-World Mathhttp://www.youtube.com/watch?v=jRMVjHjYB6w
Helping Students Develop Process Skills
CAMT Conference June 24, 2015
Dan Meyer on Real-World MathList some characteristics of his activities!
Use perplexing real-world phenomenon to make students think
Use a video instead of words to present the problem
Tap on students’ curiosity and “put students in a place where
they feel an intellectual need for new mathematical tools”
(Dan Meyer’s Blog)
Empower students with tools to answer the question
“Let math serve the conversation, … prediction, and betting,
and arguing, and not the other way around. We don’t do all that
to serve the math.” (Dan Meyer’s video on Real-World Math)
Helping Students Develop Process Skills
CAMT Conference June 24, 2015
Dan Meyer on Real-World MathWill such video-initiated activities help students
gain deep conceptual understanding and develop
mathematical process skills? Why, or why not?
2 minutes
Now!
Helping Students Develop Process Skills
CAMT Conference June 24, 2015
Dan Meyer on Real-World Math
Are such activities realistic?
Why, or why not?
Now!
Helping Students Develop Process Skills
CAMT Conference June 24, 2015
Dan Meyer on Real-World Math
Teacher-Lecture Students-Listen
Helping Students Develop Process Skills
CAMT Conference June 24, 2015
Dan Meyer on Real-World Math
Teacher-Lecture Students-Listen
Inquiry Based Learning
Helping Students Develop Process Skills
CAMT Conference June 24, 2015
Dan Meyer on Real-World Math
Teacher-Lecture Students-Listen
Task Based: Thinking & Learning
Inquiry Based Learning
What’s The Essence?
CAMT Conference June 24, 2015
Task Based: Thinking & Learning
What’s the essence underlying the process standards?
Student Active Thinking
How?
What’s The Essence?
CAMT Conference June 24, 2015
Task Based: Thinking & Learning
What’s the essence underlying the process standards?
Student Active Thinking
How?
What need to change?
Use activities that make students think!
Activities that Make Students Think
CAMT Conference June 24, 2015
ACTIVITIES THAT MAKE STUDENTS THINK
Harel’sDuality Principle
PROCESS STANDARDS
Mathematical
Habits of Mind
CONTENT STANDARDS
Interconnected
Knowledge of
Mathematics
Activities that Make Students Think
CAMT Conference June 24, 2015
An activity forstudents to experience
composite functions meaningfully
TEKS: PreCalculus Process Skills
• Problem solving
• Sense making
• Representing and
symbolizing
• Making connections
2A use the composition of two
functions to model and solve
real-world problems;
2B demonstrate that function
composition is not always
commutative
How would you use function notation to obtain your answer?
Task 1
Task 1
How would you use function notation to obtain your answer?
Task 1
Activities that Make Students Think
CAMT Conference June 24, 2015
Math Concepts
Magic-Box Task
• Strategic guessing-and-
checking
• Thinking about effect of
tripling, squaring, and
adding 5
• A composite function
concatenates two functions s.t.
the output of one becomes the
input of the other.
• Order matters in composition
of functions.
Process Skills
Students get to think
Thinking leads to the key ideas
Formalize the ideas
Activities that Make Students Think
CAMT Conference June 24, 2015
Characteristics of this Magic-Box Task
Vocabulary comes last, after students have an experience of the
meaning of the terms
The instructional sequence involves
specifying the TEKS or the math concepts
identifying key ideas underlying those concepts
designing a task that makes students think and experience those key
ideas
implementing the task where students do the thinking with minimal
teacher telling and hinting
orchestrating productive discussion
connecting the key ideas and experiences into formal symbolism
Dan Meyer on Real-
World Math
Task Based:
Thinking & Learning
Activities that Make Students Think
• Less work for teacher • More exciting for students
• Need to ensure that students
interpret the problem
situation correctly
• Both activities are designed specifically for certain math concepts
• Video enhances students’
comprehension of the problem
situation
• Use pictures and contexts to
support thinking
• Use advanced technological tools
to support student thinking
• Both activities require student to think
• Problem →Thinking → Solution →Math Concepts
CAMT Conference June 24, 2015
Changing the Way We Teach
CAMT Conference June 24, 2015
Position Yourself on This Continuum
Teacher
Showing
& Telling
Students
Thinking
with Tasks
Inquiry-
Based
Learning
Changing the Way We Teach
CAMT Conference June 24, 2015
Are You Game For A Change?
Teacher
Showing
& Telling
Students
Thinking
with Tasks
Inquiry-
Based
Learning
Changing the Way We Teach!
CAMT Conference June 24, 2015
Is it possible to change?
Our conviction
“What I show is not necessarily what students see!
What I tell is not necessarily what they hear!
What I teach is not necessarily what they learn!
What they seem to understand from me is not necessarily what they remember later!”
Our belief“Students learn best when they are cognitively engaged as opposed topassively watching and listening.”
Our desire“I want my students to THINK”.
Suppose we answer “yes”. What needs to change?
Do I really want to?
Changing the Way We Teach!
CAMT Conference June 24, 2015
What support do I need?
Our school administratorso Believe that teaching to the test is NOT effective
A professional learning communityo Form a team of like-minded colleagues
oCollaborate in finding and creating tasks that make students think
o Establish a trusting and open environment
“The definition of insanity is doing the same thing over and over again and expecting different results.” Albert Einstein
Changing the Way We Teach!
CAMT Conference June 24, 2015
What support do I need?
A professional learning communityo Form a team of like-minded colleagues
oCollaborate in finding and creating tasks that make students think
o Establish a trusting and open environment
Our school administratorso Believe that teaching to the test is NOT effective.
o Believe in long-term investment in process skills as opposed to
short-tem achievement in high-stake exams.
External support
Changing the Way We Teach!
CAMT Conference June 24, 2015
External Support
National Council of Teachers of Mathematics (NCTM)
& TCTM
Professional Development Projectso Teacher Quality Projects
o Research projects
UTEP Graduate ProgramsoMaster of Arts in Teaching - Mathematics
oMaster of Education – Instructional Specialists (Math Emphasis)
o PhD Program –Teaching, Learning, and Culture (Math Emphasis)
Concluding Remarks
Become familiar with Texas College and Career
Readiness Standards, NCTM Process Standards, CCSSM
Mathematical Practices Standards, and TEKS Process
Standards
Teach in a way to help our students develop
mathematical process skills
The essence underlying all these process standards is
student active thinking
If I really want to change, I will …
CAMT Conference June 24, 2015
Helping Students Develop
Mathematical Process Skills, Really?
CAMT Conference June 24, 2015
Thank You