Implementing Promising Implementing Promising Practices - doing the Practices - doing the

Right Thing with Right Thing with the Right Stuffthe Right Stuff

Rob Kimball

Project DirectorThe Right Stuff: Appropriate Mathematics for All Students (NSF)

Previously, on the Right Stuff…

The RIGHT Message

The RIGHT philosophy:

How do you know when learning has taken place ?

When changes in the student’s way of thinking and/or habits of mind

are altered.

What changes do we want to cause/observe

in our students?

Undergraduate programs and Courses in the Mathematical Sciences: CUPM Curriculum Guide, 2004 (p. 27)

http://www.maa.org/CUPM/curr_guide.html

The RIGHT Goals (CUPM Curr. Guide):

Offer suitable courses . . . designed to

• Engage students in a meaningful and positive intellectual experience;• Increase quantitative and logical reasoning abilities needed for informed citizenship / workplace;• Strengthen quantitative and mathematical abilities that will be useful to students in other disciplines;• Improve every student’s ability to communicate quantitative ideas orally and in writing;• Encourage students to take at least one additional course in the mathematical sciences.

Teachers who use promising practices:

• Believe knowledge is constructed – not received.

Teachers who use promising practices:

• Lead students to mental discourse and then help them resolve the issues: Elicit – Confront – Resolve – Assess

Teachers who use promising practices:

• Ask the right questions until students learn to ask them themselves. If students can’t learn to judge the quality of their own work – they really haven’t learned.

Teachers who use promising practices:

• Create a natural critical learning environment.

Finally:

Mental Models Change Slowly

A business runs an advertisement in a local paper every Thursday. The ad is 1.75 inches (height) by 2.75 inches (width) and costs $300.

However, next month, the business is celebrating 10 years of service and is prepared to spend at least twice as much for a larger advertisement. The advertising manager has the following restrictions placed on ads in the paper: all ads are measured in 1/32 of an inch the width of an ad is limited to 8/32 of an inch intervals the paper gives a 10% discount on ads that are priced over $500.

In addition, the advertising manager wants the advertisement to stay as close as possible to the golden ratio - so that it looks good to the eye.

Find the size of the ad that the manager should use and be prepared to support your conclusion.

Teaching Hint: Have students write a two-Minute paper at the end of this class: Describe how your group solved this problem and any disagreements you had

How do we want students to attack this problem ?

Module 7

Hurricanes

Hurricanes – this will blow you away

Module 7

This project is sponsored, in part, by a grant from the National Science Foundation: NSF DUE 06 32883.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Goal of the Module

Students will construct a graph and a regression model which shows the relationship between wind speed and the force of the wind. The students will analyze the math model that fits the data to make predictions from the model.

MAA CompetenciesProblem Solving

◘ solving problems presented in the context of real world situations with emphasis on model creation and interpretation

◘ creating, interpreting, and revising models and solutions of problems

MAA CompetenciesFunctions and Equations

◘ effectively using multiple perspectives (symbolic, numeric, graphic, and verbal) to explore elementary functions

◘ investigating linear, exponential, power, polynomial, logarithmic, and periodic functions, as appropriate

MAA CompetenciesData Analysis

◘ collecting (in scientific discovery or activities, or from the Internet, textbooks, or periodicals), displaying, summarizing, and interpreting data in various forms

◘ fitting an appropriate curve to a scatter plot and use the resulting function for prediction and analysis

CROSSROADS StandardsIntellectual Development

I – 1 Problem Solving

I – 2 Modeling

I – 3 Reasoning

I – 4 Connecting with other Disciplines

I – 5 Using Technology

I – 6 Developing Mathematical Power

I – 7 Linking Multiple Representations

CROSSROADS StandardsContent

C – 1 Number Sense

C – 2 Symbolism and Algebra

C – 3 Geometry and Measurement

C – 4 Function Sense

C – 5 Continuous and Discrete Models

C – 6 Data Analysis, Statistics, Prob.

C – 7 Deductive Proof

CROSSROADS StandardsPedagogy

P – 1 Teaching with Technology

P – 2 Active and Interactive Learning

P – 3 Making Connections

P – 4 Using Multiple Strategies

P – 5 Experiencing Mathematics

Introduction

Hurricanes make the news every fall. In recent years, the interest in hurricanes has escalated because of the devastation done by Katrina, and other powerful storms. This activity will attempt to quantify the damage winds are likely to cause.Video: http://www.sciencedaily.com/videos/2006/0503-hurricanes_inside_the_storm.htm

Preparing for the Module: QL skills

1. Atmospheric pressure is measured in force per unit area (click here to learn more).

One atmosphere is 14.7 pounds per square inch. Convert one atmosphere to kilograms per square centimeter.

(tns file)

atmosphericpressure.pdf

Preparing for the Module: QL skills

2. If one atmosphere of pressure exerts 14.7 pounds per square inch, how much pressure is exerted over one square foot?

Preparing for the Module: QL skills

3. A man weighing 200 pounds wears size 12 shoes. The surface area of the shoes is approximately 100 square inches.A woman wearing high heals who weighs 120 pounds comes in contact with only 12 square inches of surface.Compute the psf exerted by each person.If each stepped on your back, which one would hurt more?If each stood on a 2x4” plank, which one is more likely to break it?

Preparing for the Module: QL skills

4. According to USA Today, a wind speed of 39 miles per hour produces a force of 6.1 pounds per square foot.

How much force would be exerted on a door (6 ft 8 inches by 32 inches) by a wind of 39 mph?

Would you be able to hold the door against the wind ? <tns file>

Preliminary Thoughts and Opinions

• Ask participants to share their experiences with hurricanes

• Introduce the concepts of wind speed and air pressure (click here)

• Elicit student opinion (individually and then as a group) on what happens to the force exerted by the wind as the wind speed increases.

• Confront students with the graphic from the USA Today on wind speed and air pressure (click here).

Activity• Enter the data from the USA Today

graphic into Excel or a graphing calculator

• Construct a reasonable model to predict the force from the wind (psf) from wind speed (mph). <tns.file>

Questions

• Read the graphic more carefully and find the formula used by Ahren. How does that model compare with your’s ?

• You may ask students to do more research on Ahren’s model.

Applying the ModelBased on your model, ►What would be the force of the wind in psf

if the wind speed were 100 mph ? 200 mph ?

► If the force of the wind were 50 psf, what would the wind speed be ? 75 psf ?

► Make a table and calculate the change in the force of the wind for each increase of 5 mph in wind speed. Start the table at 40 mph. Describe the change in the rate of change of force.

Wind Speed Force

Rate OfChange

mph psf psf/mph

40 6.4  

45 8.1 0.34

50 10 0.38

55 12.1 0.42

60 14.4 0.46

65 16.9 0.50

70 19.6 0.54

More Difficult Questions

Examine the information from the Atlantic Oceanographic and Meteorological Laboratory (click here).

Is it feasible to find a model to predict the median damage of a hurricane based on its wind speed ?

Explain.

Is it feasible to find a model to predict the number of “extreme hurricane impacts” for any decade?Explain.

CONTENTMathematics courses and programs in the first two years of college need to develop students’ quantitative and workplace skills and actively engage them in the mathematics they will encounter outside the class-room. 

Faculty may need to teach content that is different from

what they were taught, teach more than they were taught, and teach differently than the way they were taught.  Students should understand some of the big ideas of mathematics through a curriculum, a variety of problem-solving strategies, and significant projects that examine selected topics in depth.  Students should have opportunities to demonstrate their mathematical knowledge, as well as their creativity.  BEYOND CROSSROADS, CH 6

CONTENT

Besides algebra, what other mathematics will students see in and use in their daily lives and in their profession?

Statistics

Probability

Networks – Graph Theory

Finance

Geometry – of solids and shapes

Integrating Content

Applications can often blend content from several areas. This helps to dispel the notion that mathematics is a collection of isolated topics.

CLASSROOM ASSESSMENT

Collecting information from students to measure their learning aids instruction and guides the instructor.

• feedback on lessons

• assessing important concepts

• improving the course

Teaching should be a reflective processTeaching should be a reflective process

Implementing

Evaluating

Planning

Documenting

Redefining

Teaching should be aTeaching should be a

process process

Implementing

Evaluating

Planning

Documenting

Redefining

Planning

The teachers who are teaching a course meet together as a team, usually once a week. Previously, they had collaboratively constructed the syllabus and put together materials that would supplement the course.In the weekly meetings, meetings led by the “Lead Instructor”, the team follows the agenda, established by the LI. Topics include ideas for applications to introduce topics in the coming week, methods that have worked before, and challenges in the upcoming material.They have previously identified the “Learning Objectives” that are measured in tests, projects, and exams.The team works together to write departmental tests that all instructors use.Minutes of all meetings are kept, sent to the department head, and placed on the web.

Teaching should be aTeaching should be a

process process

Implementing

Evaluating

Planning

Documenting

Redefining

Implementing

Teachers reflect on the ideas of others and on their own experiences and teach their class to the best of their ability.

Teaching should be aTeaching should be a

process process

Implementing

Evaluating

Planning

Documenting

Redefining

Evaluating

Teachers often use various means of assessing student learning.Minute papers – to measure understanding, to measure student satisfaction, to assess methodologyQuick Quizzes – (online, paper, clickers) students take a quick quiz to measure the degree of learning / understandingDialogue – using good questioning, faculty create an environment where students have to communicate mathematicallyLearning Objectives – bench marks for the course – data is kept on all sections

Teaching should be aTeaching should be a

process process

Implementing

Evaluating

Planning

Documenting

Redefining

Documenting

The minutes of all of the course team meetings are kept in one file with the most recent minutes placed on top. This gives a historical record of the dialogue and provides a rationale for changes that have been made.

Teaching should be aTeaching should be a

process process

Implementing

Evaluating

Planning

Documenting

Redefining

Redefining

In addition to self-reflection, teachers may assess how well instruction met the established goals in a number of ways.

Informal Discussion – teachers talk about their class with others

Peer Review – teachers, usually teaching the same course, observe each other and offer constructive feedback

Mentor – new faculty often are provided a mentor who reviews plans, observes classes, and assesses performance

Teaching should be aTeaching should be a

process process

Implementing

Evaluating

Planning

Documenting

Redefining

Our Goal

This process of continuous improvement is also our method of checks and balances. Academic freedom is not about allowing teachers to do whatever they want. There are promising practices. There is appropriate technology. There are expectations with regard to content. Teachers, guided by their own strengths and experiences, should work together to achieve common goals. All teachers; guided by research, enabled by professional development, and supported by colleagues, can grow professionally.

Module 0Module 0The Rule The Rule of Fourof Four

Implementing the Rule of Four

Module 0This project is sponsored, in part, by a

grant from the National Science Foundation: NSF DUE 06 32883.

Any opinions, findings, and conclusions or recommendations

expressed in this material are those of the author(s) and do not

necessarily reflect the views of the National Science Foundation.

Goals of the Module

This module will provide examples of how faculty should employ the rule of four in the teaching and learning of mathematics – especially with regard to college algebra.

Contents

Part 1 Linear Models

Fuel in a generator

Part 2 Polynomial Models

Price of a mile-high snack

Gas mileage

The Rule of Four

Have participants read the documentThe Rule of FourThe Rule of Four

Discuss the rule of four and the 12 pathways and what they mean.

Discuss the final figure, showing the shift to contextually based problems that motivate the mathematics.

Example 1 – Linear Models

Electric generators are used in a variety of applications. Often they provide power for signs or equipment when normal electrical service is not available. They are also found in homes and used in case of power failure.

These units usually run on gasoline and have a relatively small tank in which the fuel is stored.

The ScenarioHave participants read The Scenario for the electric generators.

Discuss the scenario and define the problem. Have participants respond to the following questions:

Is the data sufficient?Is the data appropriate?Is the data reliable?

What would you have done differently?

Organize the data and decide how it should be presented graphically.

ResultsParticipants shouldcreate a scatter plot as the most appropriate wayof representing the data graphically. However, remind them that getting to this point with all the conversation is a critical component of building the students’ problems solving skills.

Now ask, what information does this data provide?

Generator

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400

0 1 2 3 4 5 6 7 8 9 10

Gallons of Fuel

Tim

e (m

inu

tes)

Conclusions

Have participantsfind a linear modelfor the data in whatever way they wish.

Ask what the slope of this model means.

Find the intercepts and have participants write the meaning of each intercept using mathematical terminology.

Generator

y = 43.5x - 34.5

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450

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Gallons of Fuel

Tim

e (m

inu

tes)

Reasonableness – Does the model make sense?

Identify any problems with the model?

y = 43.5x - 34.5

-100

-50

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150

200

250

300

350

400

-2 0 2 4 6 8 10

A scatter plot of the data. Adjust the line to find a good model.However, the intercepts are not at the origin; something we would expect.

Moving the intercepts closer to the origin and then

re-adjusting the line finds a “good fit” Time = 39 x (# gallons)

Back to the problemThe problem though is to find out at what rate the generator uses fuel. This data indicates that the generator uses fuel at a rate of about 40 minutes per gallon.

How do we turn this information into something that is useful?

NOTE: All too often, math teachers talk about models as if they are precise – carrying out coefficients and constants to several decimal places when in reality, all we are looking for is a reasonable, working model. Thus, rounding the rate to 40 minutes per gallon is realistic and reasonable and good enough for what we want.

Model – Gallons Used

Discuss this model and what it tells us regarding the scenario. Discuss questions like those provided.

Possible Questions:

If the team usually works for 5 hours before taking a lunch break, about how many gallons should they put into the generator to last until lunch?

The team filled the generator one morning, worked 5 hours and then took a lunch break. After lunch, they needed to complete the project and planned to work until dark or about another 6 hours. How much fuel do they need to add after lunch to last until dark?

Model - Gallons Used

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5

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15

20

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Time (hours)

Gal

lons

N(t) = 1.5 t

Example #2: Number of Super-Snacks Sold

Many airlines now offer “food for purchase.” A new airline conducted an experiment to attempt to determine the price they should charge customers for a “super-snack.”

Have students read the scenario and determine how they might represent the data.

The DataStudents may determine that a scatter plot of the data is appropriate. (below)

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Price

# S

old

Have them determine a model and then define the slope in terms of

the variables.

y = -4.0463x + 32.704

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0 1 2 3 4 5 6 7

Price

# S

old

An appropriate model seems to be

# Sold = -4(price) + 33

Thus, -4 represents the change in the number sold relative to the price; or, you sell four less snacks every time you raise the price one dollar.

However, this data doesn’t tell us anything about the real question: revenue. Discuss how to find the revenue and then represent that data with a scatter plot.

Price vs. RevenueRevenue

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Price

Re

ve

nu

e

Discuss the trend you see in this data and determine an appropriate function that might model the data.

Quadratic Model

y = -5.9264x2 + 47.121x - 27.26

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Cubic Model

Chart Title

y = -1.6988x3 + 17.016x2 - 50.605x + 104.6

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Price

Rev

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Revenue – Cubic Model

ConclusionsThe cubic model seems like a better model:

y = -1.6988x3 + 17.016x2 - 50.605x + 104.6Or

Revenue = -1.7(price)3 + 17(price)2 – 51(price) + 105

Using this model, if snacks are sold for $4.50 each, the company is likely to maximize revenue.

Have students discuss the results. In particular, discuss the results in terms of practicality. A price of $4.50 will require change be given back. Sometimes, a model provides information but the implementation might cause one to choose a less than perfect solution. Going with a price of $5 per snack makes more sense.