1.
Kien Lim
Department of Mathematical Sciences
University ofTexas at El Paso
Helping Students Develop
Mathematical Process Skills, Really?
CAMT Conference June 24, 2015

2.
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?

3.
Introduction
CAMT Conference June 24, 2015
 What are in theTexas 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?

4.
Mathematical Process Skills
CAMT Conference June 24, 2015

5.
Mathematical Process Skills
CAMT Conference June 24, 2015

6.
Mathematical Process Skills
CAMT Conference June 24, 2015
;
;
;

7.
Mathematical Process Skills
CAMT Conference June 24, 2015
; ;
1. Numbers and Operations
2. Algebra
3. Geometry
4. Measurement
5. Data analysis and Probability
1. Problem solving
2. Reasoning and Proof
3. Communication
4. Representation
5. Connections

8.
Mathematical Process Skills
CAMT Conference June 24, 2015
; ;
1. Numbers and Operations
2. Algebra
3. Geometry
4. Measurement
5. Data analysis and Probability
1. Problem solving
2. Reasoning and Proof
3. Communication
4. Representation
5. Connections
CONTENT STANDARDS PROCESS STANDARDS
Are Both Sets Necessary? Why?

9.
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

10.
Mathematical Process Skills
CAMT Conference June 24, 2015
CONTENT STANDARDS PROCESS STANDARDS
Interconnected Knowledge
of Mathematics
Mathematical
Habits of Mind
Mathematics as a Discipline

11.
Mathematical Process Skills
CAMT Conference June 24, 2015
CONTENT STANDARDS PROCESS STANDARDS
Interconnected Knowledge
of Mathematics
Mathematical
Habits of Mind
IsTexas really trying to focus on
process standards?

12.
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

13.
Mathematical Process Skills
CAMT Conference June 24, 2015
Common Core State Standards
for Mathematics
PROCESS STANDARDS
TCCRS

14.
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:

15.
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.

16.
Mathematical Process Skills
CAMT Conference June 24, 2015
Common Core State Standards
for Mathematics
PROCESS STANDARDS
1. Problem solving
2. Reasoning and Proof
3. Communication
4. Representation
5. Connections

17.
Mathematical Process Skills
CAMT Conference June 24, 2015
PROCESS STANDARDS
Which set is easier to teach? Why?
Mathematical
Habits of Mind
Interconnected Knowledge
of Mathematics
CONTENT STANDARDS

18.
Helping Students Develop Process Skills
CAMT Conference June 24, 2015
1 minute
How can we help students develop
mathematical process skills?
2 minutes
Now!

19.
AYou-TubeVideo Break
Helping Students Develop Process Skills
CAMT Conference June 24, 2015
Dan Meyer on Real-World Math
http://www.youtube.com/watch?v=jRMVjHjYB6w

20.
Helping Students Develop Process Skills
CAMT Conference June 24, 2015
Dan Meyer on Real-World Math
List 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)

21.
Helping Students Develop Process Skills
CAMT Conference June 24, 2015
Dan Meyer on Real-World Math
Will such video-initiated activities help students
gain deep conceptual understanding and develop
mathematical process skills? Why, or why not?
2 minutes
Now!

22.
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!

23.
Helping Students Develop Process Skills
CAMT Conference June 24, 2015
Dan Meyer on Real-World Math
Teacher-Lecture Students-Listen

24.
Helping Students Develop Process Skills
CAMT Conference June 24, 2015
Dan Meyer on Real-World Math
Teacher-Lecture Students-Listen
Inquiry Based Learning

25.
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

26.
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?

27.
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!

28.
Activities that Make Students Think
CAMT Conference June 24, 2015
ACTIVITIES THAT
MAKE STUDENTS THINK
Harel’s
Duality Principle
PROCESS STANDARDS
Mathematical
Habits of Mind
CONTENT STANDARDS
Interconnected
Knowledge of
Mathematics

29.
Activities that Make Students Think
CAMT Conference June 24, 2015
An activity for
students 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

30.
How would you use function notation to obtain your answer?
Task 1

31.
Task 1
How would you use function notation to obtain your answer?

32.
Task 1

33.
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

34.
Activities that Make Students Think
CAMT Conference June 24, 2015
Characteristics of this Magic-BoxTask
 Vocabulary comes last, after students have an experience of the
meaning of the terms
 The instructional sequence involves
 specifying theTEKS 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

35.
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

36.
Changing the Way We Teach
CAMT Conference June 24, 2015
PositionYourself onThis Continuum
Teacher
Showing
&Telling
Students
Thinking
withTasks
Inquiry-
Based
Learning

37.
Changing the Way We Teach
CAMT Conference June 24, 2015
AreYou Game For A Change?
Teacher
Showing
&Telling
Students
Thinking
withTasks
Inquiry-
Based
Learning

38.
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 to
passively watching and listening.”
 Our desire
“I want my students toTHINK”.
Suppose we answer “yes”. What needs to change?
Do I really want to?

39.
Changing the Way We Teach!
CAMT Conference June 24, 2015
What support do I need?
 Our school administrators
o Believe that teaching to the test is NOT effective
 A professional learning community
o Form a team of like-minded colleagues
o Collaborate 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

40.
Changing the Way We Teach!
CAMT Conference June 24, 2015
What support do I need?
 A professional learning community
o Form a team of like-minded colleagues
o Collaborate in finding and creating tasks that make students think
o Establish a trusting and open environment
 Our school administrators
o 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

41.
Changing the Way We Teach!
CAMT Conference June 24, 2015
External Support
 National Council ofTeachers of Mathematics (NCTM)
&TCTM
 Professional Development Projects
o Teacher Quality Projects
o Research projects
 UTEP Graduate Programs
o Master ofArts inTeaching - Mathematics
o Master of Education – Instructional Specialists (Math Emphasis)
o PhD Program –Teaching, Learning, and Culture (Math Emphasis)

42.
Concluding Remarks
 Become familiar withTexas College and Career
Readiness Standards, NCTM Process Standards, CCSSM
Mathematical Practices Standards, andTEKS 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

43.
Helping Students Develop
Mathematical Process Skills, Really?
CAMT Conference June 24, 2015
Thank You