nationellt centrum för matematikutbildning vid göteborgs universitet teaching math through problem...

1

Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet

Teaching Math through Problem Solving

Teaching School Mathematics through Problem Solving

Diana V. Lambdin([email protected])

Frank K. Lester, Jr.([email protected])

Indiana University

2



The world is different!

• Information age

• Global economies

• New skills and concepts are needed

The mathematics curriculum must prepare students for this new world.

This means that what we teach and how we teach must change!

3



What does it mean to understand mathematics?

• Do well on national and international assessments?

• Recall facts and procedures from memory?• Solve story problems?• Apply what is learned in school in everyday

situations?• IS THERE SOMETHING ELSE?

4



Students understand a mathematical concept or skill when they:

• Demonstrate reasonable proficiency in using

it. (proficiency)

• Use it to solve problems in a variety of

contexts (and not only in school) (application)

• Know how it relates to other mathematical

ideas (connection)

5



A classroom scenario:• Grade 7• Measurement unit on area &perimeter• Teacher goes over last night’shomework• Teacher reviews definitions &formulas • Teacher shows how to find area& perimeter

7 cm.

15 cm.

50 m.

50 m.• Students begin work on a set of 20 “problems”

6



Homework

• Complete work on the 20 problems• Solve the following word problem:

A shop owner wants to put a new wood floor in his shop. The shop is 12 meters long and 8 meters wide. How many square meters of floor will the wood cover?

[Adapted from Stein et al., Implementing Standards-based Mathematics Instruction. New York: Teachers College Press, 2000]

7



A different sort of classroom• Grade 7

• Measurement unit on area & perimeter

• Students begin work on a challenging task in small groups

• Work continues for the entire class period using whatever tools the students need

• Teacher goes from group to group, asks questions, gives hints, but does not give answers

8



The rabbit pen problemMs. Gomez’s class will raise rabbits for their

spring science fair. They have 24 meters of fencing to build a rectangular rabbit pen for the rabbits.

1. If Ms. Gomez’s students want their rabbits to have as much room as possible, how long would each side of the pen be?

2. How long would each side be if they had only 16 meters of fencing?

3. How would you determine the pen with the most room for any amount of fencing? Organize your work so that someone else who reads it will understand it.

9



Another example: The cylinders problem

LAUNCH: Do cylinders with the same surface area have the same volume?

1010



EXPLORE

• Will all the Will all the cylinders hold the same hold the same amount? Explain your reasoning.amount? Explain your reasoning.

• How does changing the height affect the How does changing the height affect the circumference? Explain.circumference? Explain.

• How does changing the circumference How does changing the circumference affect the volume? Explain.affect the volume? Explain.

Questions for students:Questions for students:

1111



SUMMARIZE Have students report about their findings.Have students report about their findings. Encourage student-to-student questions.Encourage student-to-student questions. Look back: How is this problem related to problems Look back: How is this problem related to problems

we have done before? we have done before? What have we learned about the relationship between What have we learned about the relationship between

circumference and volume?circumference and volume? Examine the formulas for surface area and volume Examine the formulas for surface area and volume

(Big math ideas)(Big math ideas)SA = (2π)R*H; V = πRSA = (2π)R*H; V = πR22*H*H

1212



Extending the Activity

• Have students conjecture about what is happening to the volume as the cylinder continues to be cut, getting shorter and shorter (and wider and wider).

• Some students may become interested in exploring the limit of the process of continuing to cut the cylinders in half and forming new ones.

• What if the cylinders have a top and bottom??

1313



Qualities of the Lesson

• A question is posed about an important mathematics A question is posed about an important mathematics concept.concept.

• Students make conjectures about the problem.Students make conjectures about the problem.• Students investigate and use mathematics to make Students investigate and use mathematics to make

sense of the problem.sense of the problem.• The teacher guides the investigation through The teacher guides the investigation through

questions, discussions, and instruction.questions, discussions, and instruction.• Students expect to make sense of the problem.Students expect to make sense of the problem.• Students apply their understanding to another Students apply their understanding to another

problem or task involving these concepts.problem or task involving these concepts.

1414

The Teacher’s Primary Role in TMTPS

Coach, facilitator, designer, listener and Coach, facilitator, designer, listener and observerobserver

BUT NOTBUT NOT

Lecturer, “expert” (source of the best way to Lecturer, “expert” (source of the best way to do math)do math)



1515



Teaching Actions1.1. Motivate interest in the Motivate interest in the

problemproblem

2.2. Read the problem with the Read the problem with the students and discuss what the students and discuss what the information tells youinformation tells you

3.3. Ask: “Is this problem like Ask: “Is this problem like others you have solved?”others you have solved?”

4.4. (Optional) Discuss possible (Optional) Discuss possible strategiesstrategies

5.5. Observe and question Observe and question students to determine what students to determine what they are thinkingthey are thinking

6.6. Provide hints as needed Provide hints as needed (focus or direction hints)(focus or direction hints)

7.7. Give early solvers a Give early solvers a challenge challenge

8.8. Require all students to Require all students to “answer the question”“answer the question”

9.9. Discuss solution strategies Discuss solution strategies with entire class (Have with entire class (Have students evaluate their own students evaluate their own work)work)

10.10. Look back and look ahead Look back and look ahead (What have we learned? Can (What have we learned? Can we generalize? . . .)we generalize? . . .)

1616

Parts of a problem-solving lesson

LaunchLaunch (Actions 1 - 4) (Actions 1 - 4)

ExploreExplore (Actions 5 - 8) (Actions 5 - 8)

Summarize Summarize (Actions 9 - 10)(Actions 9 - 10)



1717

Break into small groups



1818



Beliefs about Teaching through Problem Solving: Problem solving can serve as a Problem solving can serve as a vehiclevehicle for learning new for learning new

mathematical ideas and skills. mathematical ideas and skills. A problem-centered approach to teaching mathematics A problem-centered approach to teaching mathematics

uses interesting and well-selected problems to uses interesting and well-selected problems to launchlaunch mathematical lessons and engage students. mathematical lessons and engage students.

Good problems can Good problems can inspire inspire exploration of important exploration of important math, math, nurture nurture persistence, & persistence, & reinforce the need to reinforce the need to understandunderstand strategies, math properties and strategies, math properties and relationships.relationships.

1919



And also -- PS ability develops over time and through PS ability develops over time and through

prolonged involvementprolonged involvement Math concepts, processes, and procedures Math concepts, processes, and procedures

interact and develop concurrentlyinteract and develop concurrently Instruction must be systematic, organized, and Instruction must be systematic, organized, and

teacher has a different roleteacher has a different role Content is organized around “big ideas”Content is organized around “big ideas” Lessons often involve group work, reflective Lessons often involve group work, reflective

writing, alternative assessmentwriting, alternative assessment

20



New roles and new expectations

• New roles for teachers

• New expectations for students

• Different kinds of tasks

• New emphases during instruction

• New types of assessment

2121



Challenges to teaching through problem solving

• Time

• Need for support

• Curriculum

• Alignment with assessment practices

• Unfamiliar role for (some) teachers

2222



Benefits of teaching through problem solving

• Promotes understanding• Is motivating• Helps memory• Enhances transfer• Influences attitudes and beliefs• Promotes the development of autonomous

learners

nationellt centrum för matematikutbildning vid göteborgs universitet teaching math through problem...

Documents