Motivating students: to achieve higher-order thinking skills through problem solvingAuthor(s): Mary L. Giannetto and Lynda VincentSource: The Mathematics Teacher, Vol. 95, No. 9 (DECEMBER 2002), pp. 718-723Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/20871199 .

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CHARING TEACHING IDEAS

MOTIVATING STUDENTS TO ACHIEVE HIGHER-ORDER THINKING SKILLS THROUGH PROBLEM SOLVING

Thematic units of

study pique students9

curiosity about the

integrated uses of math

ematics and

science

Many high school teachers have concerns about their students' ability to apply mathematical skills to other disciplines and situations. Some teachers believe that students should not "learn math in a

vacuum." One way to enhance students' under

standing of the concepts learned in mathematics class is to apply mathematical skills to other sub

ject areas, especially science and technology. As educators, we need to ask ourselves the fol

lowing questions:

How can we motivate high school students to ap

ply their mathematical skills to other disciplines? How can we help students become independent problem solvers?

How can we motivate students who have previ ously been unsuccessful in mathematics and science?

The authors' goal in addressing these needs is to

target high school students who have taken the minimum number of required mathematics and sci ence courses at the lowest possible level so that

they can graduate. We find that we can motivate students to learn and apply their skills in mathe

matics and science by developing an integrated mathematics, science, and technology (MST) course that is based entirely on problem solving. The MST course uses a unique approach to learning: themat ic units of study pique students' curiosity about the

integrated uses of mathematics and science. The

prerequisites for this course are one year of high school algebra and one year of any high school labo

ratory science. The course is taught in an extended block of time. An eighty- to ninety-minute class

period is ideal. In this article, we present parts of a single unit

of study on flight. Flight interests many students and furnishes a natural context for problem-solving activities that involve mathematics and physics. The objectives of this unit are the following:

Students begin to understand the basic concepts of flight. Students investigate and research natural and man-made flight designs.

Students investigate Newton's laws of motion and

apply them to specific problem-based activities.

Students design and build a working model to

apply the principles of projectile motion.

Students design and build a working model to

apply Bernoulli's principles of aerodynamics.

Each thematic unit of study contains eight to ten hands-on problem-solving activities. The first activ

ity is a research project; in the example of the flight unit, students investigate and compare methods of

propulsion and flight. Students first work individu

ally on their research, and then they compare their results in pairs. The student pairs then communi cate their findings to the entire class. An example of a research project is comparing the flight of a

hang glider with that of a flying squirrel. The class

generates topics in a student-centered brainstorm

Photograph by Lynda Vincent, all rights reserved

"Sharing Teaching Ideas" offers practical tips on teaching topics related to the secondary school mathematics cur

riculum. We hope to include classroom-tested approaches that offer new slants on familiar subjects for the begin ning and the experienced teacher. Of particular interest are alternative forms of classroom assessment. See the

masthead page for details on submitting manuscripts for review.

718 MATHEMATICS TEACHER

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Photograph by Lynda Vincent, all rights reserved

ing activity, which may be supplemented by sugges tions from the teacher. A list of ideas generated by our students includes helicopters, bumblebees, dragonflies, gliders, kites, flying squirrels, and bats. Students use the Internet to research their topics.

The students' favorite activity in the flight unit involves parabolic motion. This activity takes place around the middle of the school year. Students have

already studied trigonometry and parametric equa tions. The students have also had some experience with graphing calculators. Working in pairs, stu dents design a golf-ball launcher (see sheet 1). Stu dents use formulas to calculate the answers to

problems 1, 2, and 3 on sheet 2, then they apply the techniques that they have practiced to make

predictions about the velocity, distance, and angle measure of their launchers. They can also use para metric equations to make predictions. They use their device to calculate the initial velocity of their launcher. Then, using the calculated initial velocity, parametric equations, and a graphing calculator, students predict the horizontal distance that their

golf ball will travel to hit a target, given a specific angle, and calculate the launch angle necessary to hit a target at a specified horizontal distance from the launcher. As with all the activities in each the matic unit, the teacher uses a series of rubrics for assessment. Figure 1 shows the grading policy for sheet 1. The design of the launcher and the list of materials are generated entirely by the students.

Completing this activity not only improves stu dents' mathematics and science skills but also

helps them learn about the other forces that affect the flight of their projectile. Students soon discover that the flight of the golf ball is affected by such variables as wind, barometric pressure, changes in

temperature that affect the elasticity of rubber, and the dimpling of the golf ball's surface. Students enhance their technology skills by using parametric

equations and interpreting graphs on their graph ing calculators. Self-assessment allows students to reflect on the reasons for their successes or failures, and they can improve their designs.

This activity and many others like it motivate

students, especially those who would normally have taken only the most basic mathematics and science courses needed to graduate. Through their success es in this course, many students who are not on a

college-bound track choose not only to take more

challenging mathematics and science courses but have also continued their education at a post secondary school.

This course is in its seventh year of existence. We limit the class size to no more than twenty stu dents. The passing rate has been higher than 85

percent each year. The enthusiasm of students in

Self assessment

allows

students to

improve their

designs

GRADING POLICY FOR SHEET J Parabolic motion

Design a) Functional b) Safe c) An approved drawing of the design

Operation of launcher and hitting the target area Lab writeup

25%

25% 50%

Grading rubric

90-100 1. Mathematical, scientific and technology terms are correctly used.

2. All calculations are correct. 3. All graphs, charts, designs, and so on are correct. 4. Sequence of events is communicated correctly. 5. Error analysis is correct.

6. Conclusion is logical. 7. The student has applied what he or she learned to other areas

(extensions). 8. A reflection about the activity is included.

80-69 1. Mathematical, scientific, and technological terms are correctly used.

2. All calculations are correct. 3. All graphs, charts, designs, and so on are correct. 4. Sequence of events is communicated in a somewhat logical

order.

5. Error analysis is mostly correct. 6. The work shows a conclusion. 7. A reflection about the activity is included.

70-79 1. Terminology is used. 2. Calculations are included. 3. Graphs, charts, and so on are used.

4. Sequence of events is communicated. 5. An error analysis is included. 6. A conclusion is present.

60-69 1. Terminology is incorrectly used. 2. Few calculations are included, or calculations are not correct. 3. No graphs, charts, and so on are included. 4. No error analysis is included. 5. Conclusion is incorrect.

Fig. 1

Vol. 95, No. 9 December 2002 719

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the class ensures a full class the following year. At the end of each year, we ask the students to write an evaluation of the course. Students are over

whelmingly positive and often suggest new activi ties. The course is always a work in progress. We continue to supplement and change activities

according to the needs of our students. Anyone interested in learning more about our course can

contact the authors.

SOLUTIONS 1. a) initial velocity = 80.4 ft./sec.

b) initial velocity = 98.04 ft./sec.

c) initial velocity = 131.64 ft./sec.

2. a) time = 2.8 sec; distance = 154.45 ft.

b) time = 1.7 sec; distance = 111.1 ft.

c) time = 3.5 sec; distance = 150.8 ft.

3. a) 19.1 degrees b) 9.4 degrees

Students' actual results may vary because of

wind, air temperature, elasticity, and so on. There

fore, calculation is a starting point to predict actual distances.

RESOURCES TI Cares magazine www.nasa.gov

www.exploratorium.edu www.sciam.com

www.ti.com

The authors would like to acknowledge their col

leagues who collaborated with them in writing this course. They are Chris Rogine, of Pawling High School, Pawling, NY 12564; and Craig Trachten

berg and Eric Harvey, of Webutuck High School, Amenia, NY 12501.

^^ j Mary L. Giannetto

^^^BH^H giannettom@northsalem .k.l2.ny.us

Lynda Vincent

Lyndav53@aol. com

North Salem High School North Salem, NY 10560

Mr

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720 MATHEMATICS TEACHER

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FLIGHT UNIT: SUMMARY OF ASSIGNMENT SHEET 1

Objective: To design a launching device that propels a golf ball to hit a target area

Suggested materials: Wood, target, dowels, piece of leather, bungee cords, screws, hinges, nails,

protractors, graphing calculator, glue, golf ball

Construction:

1. Students draw a design of a launching device; the teacher must approve the design.

2. Students generate a list of materials needed for their design. 3. After the teacher approves the design, students build their launching devices.

Procedure:

1. Students perform trials to determine launch angle for maximum horizontal distance.

2. Students use the determined launch angle to calculate initial velocity.

3. Students are given specified angles, and they compute the horizontal distance that the projec tile will travel. Students may take two or three trials to prove that their calculations are correct.

4. Students are given specified distances to the target area, and they compute the angles neces

sary to hit the target. Students may take two or three trials to prove that their calculations are

correct.

From the Mathematics Teacher, December 2002

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PARABOLIC MOTION SHEET 2A

This worksheet familiarizes you with the equations that are necessary to calculate the initial veloci ty of your launching device. When a projectile is launched at an angle from the horizontal, two forces act on the body that is

in motion. One is the horizontal force, and the other is the vertical force. The necessary equations are as follows:

sx = v0t cos 0,

the distance along the horizontal axis, and

sy = -^gt2 + v0t sin 6 + h,

the distance along the vertical axis raised some height h above the ground, where sx is the hori zontal displacement (distance); sy is the vertical displacement; v0 is the initial velocity; t is the time of flight; 0 is the launch angle; g is the gravitational constant 32 ft./sec, or 9.8 m/sec; and h is the height above the ground of the launching device.

To start, let the launch angle 0 equal 0?. Since cos 0? = 1 and sin 0? = 0, the previously given equations simplify to sx = v0 t and sy

= -{M2)gt2 + h.

You can find the initial velocity of an object if you know the horizontal distance that it travels and the time taken to travel that distance. Since the time taken to travel a certain distance is the same as falling from a given height, we can use the formula

o = -\gf

+ h.

Once you have the time and know the horizontal distance traveled, you can use the formula sx = v0t cos 0.

1. Calculate the initial velocity in each of the following examples:

a) height = 5 feet, and horizontal distance = 45 feet

b) height = 1.6 feet, and horizontal distance = 31 feet

c) height = 12 feet, and horizontal distance = 114 feet

From the Mathematics Teacher, December 2002

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PARABOLIC MOTION-Continued SHEET2B

After you know the initial velocity of your launcher, you can calculate the horizontal distance

that your projectile will travel for a given angle by using the formulas

0 = -^gt2

+ v0t s\n 6 + h

and

sx =

v0t cos 0.

2. In the following examples, let the initial velocity, v0, be 70 feet/sec, and let the height be 5. Com

pute the time and horizontal distance that the projectile will travel.

a) 0 = 38 degrees

b) 0 = 21 degrees

c) 6 = 52 degrees

Hint: use the formula

0 = -Igt2 + v0t sin 6 + h \gt2

to find time, and then use sx = v0t cos 0to find the horizontal distance.

3. You can use the equation

6 = sin -1 ? s X

to find the angle if you know the horizontal distance that the projectile traveled and its initial

velocity. Find the angle for each of the following:

a) velocity = 75 feet/second, and sx = 115 feet

b) velocity = 42 feet/second, and sx = 36 feet

From the Mathematics Teacher, December 2002

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