Helping Students Develop Mathematical Process Skills, Really?

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

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

1. Numbers and Operations2. Algebra3. Geometry4. Measurement5. Data analysis and Probability

1. Problem solving2. Reasoning and Proof3. Communication4. Representation5. Connections

1. Numbers and Operations2. Algebra3. Geometry4. Measurement5. Data analysis and Probability

CONTENT STANDARDS PROCESS STANDARDS

Are Both Sets Necessary? Why?

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

Interconnected Knowledge

of Mathematics

Mathematical

Habits of Mind

Mathematics as a Discipline

Interconnected Knowledge

of Mathematics

Mathematical

Habits of Mind

Is Texas really trying to focus on

process standards?

Revised TEKS – High School Math PROCESS STANDARDS

1. Problem solving

2. Reasoning and Proof

3. Communication

4. Representation

5. Connections

Common Core State Standards for Mathematics

PROCESS STANDARDS

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:

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.

PROCESS STANDARDS

Which set is easier to teach? Why?

Mathematical

Habits of MindInterconnected Knowledge

of Mathematics

CONTENT STANDARDS

Helping Students Develop Process Skills

1 minute

How can we help students develop

mathematical process skills?

2 minutes

A You-Tube Video Break

Dan Meyer on Real-World Mathhttp://www.youtube.com/watch?v=jRMVjHjYB6w

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)

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

Dan Meyer on Real-World Math

Are such activities realistic?

Why, or why not?

Teacher-Lecture Students-Listen

Inquiry Based Learning

Task Based: Thinking & Learning

Inquiry Based Learning

What’s The Essence?

What’s the essence underlying the process standards?

Student Active Thinking

What’s The Essence?

What’s the essence underlying the process standards?

Student Active Thinking

What need to change?

Use activities that make students think!

Activities that Make Students Think

ACTIVITIES THAT MAKE STUDENTS THINK

Harel’sDuality Principle

PROCESS STANDARDS

Mathematical

Habits of Mind

CONTENT STANDARDS

Interconnected

Knowledge of

Mathematics

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

How would you use function notation to obtain your answer?

Task 1

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

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

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

• 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

Changing the Way We Teach

Position Yourself on This Continuum

Teacher

Showing

& Telling

Students

Thinking

with Tasks

Inquiry-

Learning

Changing the Way We Teach

Are You Game For A Change?

Teacher

Showing

& Telling

Students

Thinking

with Tasks

Inquiry-

Learning

Changing the Way We Teach!

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?

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

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

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 …

Helping Students Develop

Mathematical Process Skills, Really?

Thank You

Helping Students Develop Mathematical Process Skills, Really?

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