Using Mathematical Tasks To Provoke Intellectual Needs

Kien LimUniversity of Texas at El Paso

kienlim@utep.edu

57th Annual Conference for the Advancement of Mathematics TeachingJuly 16, 2010

• What Constitutes a Good Mathematical Task?

• What is Intellectual Need?Why is it Important?

• How does an Intellectual-need-provoking Task Facilitate Mathematical Learning?

Objectives

Sequence Of Activities

1. Find out what constitute a good mathematical task

2. Solve a few problems

3. Discuss

4. Conclude

One More Problem?

What is “Mathematics”?

Old-view Mathematics• Collection of definitions,

formulas, rules, and procedures

• The focus is on the “how” and the “answer”

• Students are taught the procedure and then practice it

New-view Mathematics• A subject that involves

thinking and sense-making• The focus includes the

“why” and meaning• Students engage in

problem-solving, making connection, justifying, representing, and communicating

Purpose of Classroom Tasks

• To engage learners in thinking, sense-making, reflecting, and abstracting

• To engender discussions among learners to learn “new” mathematical knowledge

• To provide learners an opportunity to practice what they have learned

Thompson, Carlson, and Silverman (2007)

Characteristics of a Good Task

• Mathematically driven

• Requires justifications and explanation

• Intrinsic to students

Van de Walle (2003)

Tasks that Motivate Students

• Extrinsically Motivating Something new (e.g., manipulatives) Fun and exciting (e.g., game)

• Intrinsically Motivating Intriguing

What Leads to Mathematical Learning?

“Students are most likely to learn when they see a need for what we intend to teach them, where by ‘need’ is meant intellectual need, not social or economic need.” (Harel, 1998)

“For students to learn what we intend to teach them, they must have a need for it, where by ‘need’ is meant intellectual need, not social or economic need.” (Harel, 2007)

Gremlins are rather smart.

Although they have four fingers in each hand, they are able to represent 25 different numbers, from 0 to 24, with two hands. Can you figure out how they do it?

Problem #1 for You

• Do you find the task interesting?

• Did you experience some sort of puzzlement?

• What do you think is the mathematical concept for which this task is trying to provoke?

Follow-up Questions

The line segment represents 1 2/3 km. Extend the line to represent 3 3/4 km. Be as accurate as you can without measuring the actual length of original line segment.

Problem #2 for You

1 2/3 km

• What do you think is the math concept for which this task is trying to provoke?

• What key ideas are necessary for solving this problem?

Follow-up Questions #2

1 2/3 km

1 km

How?Cut into 5 pieces.

1 km 1 km 3/4 km

Why 5?1 2/3 = 5/3

Referent unit (1km)

Mixed num. - improper fraction conversion

Consider these 4 products of numbers:

Problem #3 for You

Without computing the actual value of each product, identify those products that have the same value.

(i) 25261112 (ii) 3112544

(iii) 512255(iv) 5102233

What do you think is the math concept for which this task is trying to provoke?

Follow-up Question #3

= 24 32 52 11 = 22 3 52 112

= 24 32 52 11 = 22 3 52 112

(i) 25261112 (ii) 3112544

(iii) 512255(iv) 5102233

How Does Intellectual-Need-Provoking Task Facilitate Learning?

• Students encounter a problematic situation due to the limitation of their existing knowledge

• They experience a desire to resolve the situation

• Their resolution of the situation may lead to construction of new knowledge or modification of existing knowledge

In Selecting a Need-Provoking Task, What Questions Can We Ask Ourselves?

• What mathematical learning can the task potentially provoke?

• Is the task intrinsic to students?

• Do students experience a limitation of their existing knowledge and a need for the new math idea in order to solve the problem?

Recapitulation

What Problems Have We Worked On Today?

1. The Gremlin Problem

2. Extending 1 2/3 km Line Problem

3. Finding Equal Product Problem

What Concept do these Problems Seek to Provoke?

Place-Value

Ruferent Unit & Procedure for Converting Mixed # to Improper Fraction

Prime Factorization

Factors to Consider When Designing Need-Provoking Tasks

• The challenges associated with the math idea that we want our students to learn

• A typical student’s current knowledge

• Connection to past activities and future topics

• Classroom norms and practices

As a Math Teacher, You Can …

• Avoid problem-free tasks

• Provide opportunities for your students to experience the intellectual need for the math concepts that you want them to learn

• Select tasks that requires thinking, sense-making, exploring, justifying, and explaining

An Article

Lim, K. H. (2009). Provoking intellectual need. Mathematics Teaching in the Middle School, 15(2), 92-99.

Presenter: Kien Lim

Date: July 17, 2010 (Sat)

Time: 11:45am – 12:45pm

Venue: 101 A/B

The Hammer-and-Nail Phenomenon: Addressing

Students’ Impulsive Disposition in Mathematics

Classrooms

Thank You