APPENDIX D Sample Materials

D-1 Beliefs About Mathematics and Problem Solving

Directions: For each item below circle the letter which corresponds to the measure of your agreement or disagreement with the statement.

1. Solving a mathematical problem usually relies on knowing some mathematical facts.
A. Strongly agree B. Agree C. Undecided D. Disagree E. Strongly disagree

2. Mathematics is mostly a set of rules for doing problems.
A. Strongly agree B. Agree C. Undecided D. Disagree E. Strongly disagree

3. Much mathematics is not really useful except for people who do very specialized work.
A. Strongly agree B. Agree C. Undecided D. Disagree E. Strongly disagree

4. Either people can do mathematics or people cannot do mathematics.
A. Strongly agree B. Agree C. Undecided D. Disagree E. Strongly disagree

5. There is only one correct answer to any mathematics problem.
A. Strongly agree B. Agree C. Undecided D. Disagree E. Strongly disagree

6. If a mathematical problem can be solved, it can usually be done in five minutes or less.
A. Strongly agree B. Agree C. Undecided D. Disagree E. Strongly disagree

7. There is only one way to do a mathematics problem correctly.
A. Strongly agree B. Agree C. Undecided D. Disagree E. Strongly disagree

8. To do mathematics is to calculate answers to problems.
A. Strongly agree B. Agree C. Undecided D. Disagree E. Strongly disagree

9. Learning mathematics involves mostly memorizing.
A. Strongly agree B. Agree C. Undecided D. Disagree E. Strongly disagree

10. Solving some mathematical problems involves knowing different strategies to try.
A. Strongly agree B. Agree C. Undecided D. Disagree E. Strongly disagree

D-2 Some Problems Related to Minimal Competency

1. Liam's bowling average is 134. Every Wednesday night at his league, Liam bowls five games. If his first three games are 130, 129, and 151, then what must Liam average in his last two games to keep his average at a 134 overall?

2. Sally knows that her Grandmother's five-digit zip code in south Florida begins with either "33'' or `"34''. How many zip codes has Sally narrowed it down to?

3. A 7-ounce can of pork and beans sells for \$1.26. What is the highest price that a 20-ounce can may sell for and still be more economical than the smaller can?

4. Two neighbors, Wilma and Betty, each have a swimming pool. Both Wilma and Betty's pools hold 9000 gallons of water. If Wilma's garden hose fills at a rate of 650 gallons per hour while Betty's garden hose fills at a rate of 500 gallons per hour, then how much longer does it take Betty to fill her pool than Wilma?

5. The following appeared in a 1988 news magazine: How many alcoholic beverages did you drink in the past week? (From a telephone poll of 664 people with sampling error +/- 4%).

 0 32% 15-19 1% 1-7 49% 20-29 4% 8-14 9% 30+ 3% no opinion 3%

6. The following paragraph is from USA TODAY, March 1992:

"I wonder whether we as a society fully understand and appreciate the subtext of the \$650,000,000,000 we spend for health care? For example, how widely is it understood that total health care expenditures have risen 30\% faster than the GNP since 1970 while health care prices have increased 60% more rapidly than general price inflation? Is it fully appreciated that, despite this growth in the total amount of services, the even faster rise in their costs, and the \$2,600 per capita we now are spending, access to basic health care services remains severely restricted for much of the population?''

In a few sentences, describe what is meant by "...total health care expenditures have risen 30\% faster than the GNP since 1970...''

7. Connie has an unlimited supply of dimes and quarters. Her phone call at the public telephone is \$.25 for the first minute and \$.17 for each additional minute or portion thereof. She knows that her call will be 4 minutes and 30 seconds in length.

a. Determine the different combinations of coins that Connie can deposit to pay for her call.

b. Explain which way (or ways) is most economical for her.

8. To estimate the number of pigeons in a local park, city workers catch 82 and tag and release them. Later they catch 67 and find 20 of these tagged. How many pigeons should they expect to be in the park?

9. Jose has a refigerator with a freezer compartment which measures 26 inches by 15 inches by 15 inches. Freezer containers come in cartons of eight with each container a cube with 3.5 inches per edge. How many cartons must Jose buy to be sure he can fully utilize his freezer compartment?

10. On a loan of \$1,000 Sophie is required to pay \$45.91 for 24 monthly payments. Assuming interest is compounded monthly, what is the true annual interest rate of this loan?

11. Gloria noted that in June of 1989 her tuition to Super-U increased by 6% and in June of 1990 it increased by 3.5%. Over the two year span what was the percentage increase in tuition?

12. According to the Congressional Budget Office the actual expenditures for the federal government for the fiscal year 1992 were 1129 billion dollars. Anticipated expenditures for fiscal year 1993 are 1149 billion dollars, and the proposed federal budget for fiscal year 1994 is 1217 billion dollars. If a graph were drawn plotting expenditures versus time, would it have the appearance of a line segment connecting the point plotted for 1992 and the one plotted for 1994? Explain your answer in a complete sentence.

13. A laboratory technician has a large amount of a solution which is 40% alcohol and a large amount of a solution which is 60% alcohol. She would like to make 8 liters of a solution which is 55% alcohol. Determine the amount of 40% alcohol solution she will need and explain in a few sentences how you got your answer.

14. The Amtrack train is running 25 minutes behind schedule as it leaves Doverdale, traveling west to Topper Creek which is 78 miles away. If the train normally travels at 50 miles an hour, about how much faster would I have to travel to make up the lost time and arrive in Topper Creek on Time?

15. Harry raises tropical fish and noticed his guppy population increasing according to the following table:

 Time in weeks 0 4 8 No. of guppies 3 33 363

If the guppies continue to increase at this rate, how many should he expect to have in twelve weeks? Explain your answer.

16. Dave works at a coffee shop. A customer asks for 1 1/2 lbs blend of 1/3 vanilla creme and 2/3 decaffienated Columbian. Dave grinds a blend of 2/3 vanilla creme with 1/3 decaffeinated Columbian. How can he give the customer what she asked for while wasting the least amount of coffee? Explain your answer in a couple sentences.

15. Graduation pictures can be bought individually or in packages. Mary wants an 8 x 10, three 5 x 7's and 11 wallet size. Prices are listed below. What's the best deal for Mary? Explain your answer in a couple sentences.

 Package A Package B Individual 1 - 8 x10 2 - 5 x 7 8 x 10 \$18.00 10 wallets 12 wallets 5 x 7 \$ 6.00 \$25.00 \$20.00 wallet \$ 1.00

D-3 Project Ideas Project One: Chi-square Testing

You are to do this project working with your team of 4 students, but each student in the team is to do his/her own write-up of the project.

Your question to be considered is do male and female NIU students respond differently to the question of whether every American high school graduate should be able to complete a college degree. Determine to use the chi-square statistic to test the null hypothesis that there is no significant difference between male and female NIU students in response to Q: "Do you think every American high school graduate should be able to complete a college degree?''

Proceed as follows:

1. Determine the level of significance you (your team) will use for the test.

2. Determine how your team will obtain a good sample of 80-100 NIU students to be polled.

3. Conduct the poll of the sample asking the question Q and allowing the responses "Yes'', "No'', "No opinion''. Tabulate responses.

4. Determine the expected distribution for responses to Q assuming the null hypothesis and the responses "Yes'', "No'', "No opinion''.

5. Determine the number of degrees of freedom for the chi-square test of the null hypothesis and the value of the x2 statistic which would cause rejection of the null hypothesis.

6. Compute the x2 statistic for the data obtained and tell if the null hypothesis should be rejected.

7. Write up a short report on your study including a description of the points carried on in #1-6 and a brief paragraph discussing the findings of the study.

Project Two: Personal Business Applications (for collaborative study).

You are to do this project working with your team of 4 students, but each student in the team is to do his/her own write-up of it.

1. The Howies want to purchase a home for \$98,000 and want a 30 year fixed rate mortgage for 90% of the purchase price. Three finance companies give quotes:

 Center Bank 7% plus 1.75 points First Federal 7.375% plus no points Fleet Mortgage 7.25% plus one point

Which one should he choose and why?

2. The Jones want \$80,000 in 16 years for their son Horace to use for college. They can afford to invest \$117 monthly. What is the annual interest rate they will need to have for their investment in order to reach their goal? Explain.
Interest is assumed to be compounded monthly.

3. Sophie wants some photography equipment which totals \$1780. She makes a down payment of \$275. The balance is to be financed at 11% with interest compounded monthly.

a. What is the minimum monthly payment she must make in order to ever pay off her loan? Explain your answer.

b. What would her monthly payment be if she wanted to pay off the loan in two years? Explain.

D-4 Scoring Guides

I. An analytic scoring scale for the evaluation of problem solving capacity (an adaptation of a scale in the booklet How to Evaluate Progress in Problem Solving by Randall Charles, Frank Lester, and Phares O'Daffer published in 1987 by the National Council of Teachers of Mathematics).

Stages of problem solving Scoring scale
(from George Polya's How to Solve It)
A. Understanding the problem 0: Complete misunderstanding of the problem
1: Only part of the problem misunderstood or misinterpreted
2: Complete understanding of the problem
B. Devising a plan to solve the problem 0: No attempt, or totally inappropriate plan
1: Partially correct plan based on part of the problem being interpreted correctly
2: Plan could have led to a correct solution if implemented properly
C: Carrying out the plan 0: No answer, or wrong answer based on an inappropriate plan
1: Copying error; computational error; partial answer for a problem with multiple answers
2: Correct answer and correct label for the answer.
D: Looking back at the solution 0: No answer or answer does not make sense in the context of the problem.
1: Answer makes sense even if not correct; solution is okay but could be better.
2: Solution is correct and a good one. Argument or result is checked in other ways when it can be. Interpretation appropriate and makes sense.

II. A scoring guide for judging the accomplishment of quantitative literacy goals.

A quantitatively literate college graduate should be able to:

1. Interpret mathematical models such as formulas, graphs, tables, and schematics, and draw inferences from them.

2. Represent mathematical information symbolically, visually, numerically, and verbally.

3. Use arithmetical, algebraic, geometric and statistical methods to solve problems.

4. Estimate and check answers to mathematical problems in order to determine reasonableness, identify alternatives, and select optimal results.

5. Recognize that mathematical and statistical methods have limits.

For each of the five capabilities listed above evaluate student attainment of that capability according to the following:

Score Rating Characteristics of student work
5 Excellent Insightful understanding is effectively demonstrated. Whether directly stated or implied, responses are perceptive and appropriately supported by specific details. Support is well defined and appropriate.
4 ProficientWell-considered understanding is appropriately demonstrated. Whether directly stated or implied, responses are thoughtful and supported by details. Support is well-defined and appropriate.
3 Satisfactory A defensible understanding is clearly demonstrated. Whether directly stated or implied, responses are conventional but plausibly supported. Support is general but functional.
2 Limited Understanding is vaguely demonstrated or is not always defensible or sustained. Responses are superficial and support is scant and/or vague, and/or redundant.
1 Poor An implausible conjecture is suggested. Responses are irrelevant or incomprehensible. Support is inappropriate, inadequate, or absent.
0 Insufficient The marker can discern no evidence for the capability. Responses are so deficient that it is not possible to assess the capability.

(Adapted from a guide for assessing writing assignments which appears in the book Assessing Student Performance by Grant P. Wiggins published in 1993 by Jossey-Bass Publishers.)

D-5 Approaches to Quantitative Literacy for Some Specific Majors

I. Social psychology---A foundations finite mathematics course is followed by a sophomore-level elementary statistics course. A junior-level research methods in psychology course which presupposes the finite mathematics course and the statistics course is taken. The research methods course is followed by at least one of three laboratory courses which are senior-level-- -clinical psychology laboratory, social and personality laboratory, and developmental psychology laboratory.

II. Music---A foundations quantitative literacy course is taken in the freshman year of studies. In each of the succeeding years, sophomore, junior, and senior, one course which has a substantial quantitative component is taken in fulfillment of the general education requirements---e.g., in the sophomore year principles and problems of microeconomics; in the junior year introductory geology; and in the senior year human nutrition.

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