Plan for today

1. Plan for today • Challenges of education/assessment for QL

2. Challenges of QL Education & Assessment • QL is difficult – words & numbers, contexts, unpredictability, ill-defined problems • QL cannot be taught – habit of mind must be learned, must be practiced, productive disposition • QL seems to require immersion across the curriculum • Canonical problem situation – read, interpret, model, solve/compute/represent, reflect, & critique • Students have anti-QL habits formed in traditional mathematics classes

3. The New York Times - October 14, 2001 One big advantage… we are surrounded by sample problems… we just have to learn how to educate for solving them and to assess the resulting learning. Another advantage .. we worry about how to educate for QL… so we should rely on assessment of learning to guide our work.

4. QL Assessment Questions to Consider • What are the learning goals for QL? • How can students' progress towards QL objectives be assessed? • What are the developmental steps in QL? • What can current standardized tests tell us about students' quantitative literacy? • What should we value, i.e. what should we score? • What are the standards for proficiency? • Can we assess whether or not students are inclined to practice? • How are mathematical and numeracy skills related? • What knowledge is needed?

5. Grant Wiggins on assessing QL • Requires assessment of complex realistic, meaningful, and creative performances – authentic tasks • Authentictasks require • Construction of knowledge • Disciplined inquiry • Value beyond school • Threatens all mainstream testing and grading in all disciplines, especially mathematics

6. QL is not a new problem, just a recasting In 15th & 16th century England, according to Pat Cohen in A Calculating People, “Strange as it seems, commercial life both triggered and then limited the development of numeracy in England. The adoption of Arabic numerals and arithmetic came as a result of the expansion of commerce in the 16th century, but textbook writers then decided that their subject was too difficult for bourgeois lads to learn with any degree of understanding. They tried to strip arithmetic to its essentials, but in fact they cut it into incoherent bits and made it an arcane subject, almost impossible to learn.”

7. Canonical problem situation • Glean out the relevant information. • Have confidence to take up the challenge. • Estimate to see if assertions are reasonable. • Do the mathematics. • Generalize the situation. • Reflect on the results.

8. Mathematical Proficiencies Needed • Calculate and estimate -- decimals and fractions • Recognize and articulate mathematics • Generalize and abstract specific mathematical situations • Functions as process -- especially linear and exponential growth • Some facility with algebra • Use calculators to explore and compute • Other examples will require knowing about shape -- geometry and measurement -- and about data analysis and probability

9. Mathematical Proficiency • Conceptual understanding of mathematical concepts, operations, and relations • Procedural fluency • Strategic competence • Adaptive reasoning • Productive disposition, that is, the habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy From NRC Study Report, Adding It Up, Kilpatrick, Swafford, & Findell, 2001

10. Plan for today • Changes needed

11. Issues with Traditional Courses • Emphases on components not processes • Lack of mental constructs in lower level courses • Lack of venues for continued practice beyond the course • Not organized like the real world • Tend to degenerate to methods and procedures • Develop template exercises expectations • Not enough ambiguity • Not enough interpretation and reflection

12. Changes in Pedagogy • Mathematics should be encountered in many contexts such as political, economic, entertainment, health, historical, and scientific. Teachers will require broader knowledge of many of the contextual areas. • Pedagogy is changed from presenting abstract (finished) mathematics and then applying the mathematics to developing or calling up the mathematics after looking at contextual problems first. • Material is encountered as it is in the real world, unpredictably. Productive disposition is critical for the students. • Much of the material should be fresh -- recent and relevant.

13. Changes in Pedagogy • Considerably less mathematics content is covered thoroughly. • The mathematics used and learned is often elementary but the contexts are sophisticated. • Technology – at least graphing calculators – is used to explore, compute, and visualize. • QL topics must be encountered across the curriculum in a coordinated fashion. If I can coach writing then literature faculty can coach QL. • An interactive classroom is important. Students must engage the material and practice retrieval in multiple contexts.

14. Plan for today • Can one play the game?

15. Playing the game vs Individual skills • Dribbling • Passing • Shooting free throws • Jump shot • Know rules • Know positions • Know screens • Know defense But can you play the game?

16. Game of QL • Compute • Estimate • Graph • Interpret graphs • Solve equations • Geometry - area & volume • Rates of change • Odds & probability • Averages • Etc. • But, can you read the newspaper?

17. Processes • Reading • Problem solving • Modeling • Reasoning • Communicating • Critiquing • Reflecting • Comparing

18. Plan for today • Assessment items

19. The sales price of a car is $12,590, which is 20% off the original price. • What is the original price? • $14,310.40 $14,990.90 $15,290.70 $15,737.50 $16,935.80 • Solve the following equation for A :  2A/3 = 8 + 4A • -2.4 2.4 1.3 -1.3 0 • If r = 5 z then 15 z = 3 y, then r = • Y 2 y 5 y 10 y 15 y • If y = 3, then y3(y3-y)= • 300 459 648 999 1099

20. Mary is reviewing her algebra quiz. She has determined that one of her solutions is incorrect. Which one is it? 2x + 5 (x-1) = 9            x = 2 p - 3(p-5) = 10              p = 2.54 y + 3 y = 28               y = 4 5 w + 6 w – 3w = 64     w = 8 t – 2t – 3t = 32              t = 8 Which of the following is not a rational number? -4 1/5 0.8333333…… 0.45 Square root of 2 If Sam can do a job in 4 days that Lisa can do in 6 days and Tom can do in 2 days, how long would the job take if Sam, Lisa, and Tom worked together to complete it? 0.8 days 1.09 days 1.23 days 1.65 days 1.97 days

21. Add 0.98 + 45.102 + 32.3333 + 31 + 0.00009 368.573 210.536299 109.41539 99.9975 80.8769543 41% equals: 4.1 .41 .041 .0041 .00415 At a certain high school, the respective weights for the following subjects are Mathematics 3, English 3, History 2, Science 2 and Art 1. What is a student’s average whose marks were the following: Geometry 89, American Literature 92, American History 94, Biology 81, and Sculpture 85? 85.7 87.8 88.9 89.4 90.2 What is the median of the following list of numbers? 4, 5, 7, 9, 10, 12 6 7.5 7.8 8 9

22. A building measures 20 ft wide by 32 ft long. If it has a flat roof, which must be covered by plywood sheets measuring 4 ft by 8 ft, how many plywood sheets are needed to cover the roof? A. 32 B. 20 C. 8 D. 16 If you drove your automobile 396 miles and used 18 gallons of gasoline, what is your gas mileage in miles per gallon? A. 22 B. 20 C. 16 D. 14

23. In 2000, the National Center for Educational Statistics reported the number of public high school graduates by state for the 1995-1996 to 1999-2000 school years. The total number of high school graduates in the United States for the school year beginning in 1998 was roughly 2.5 million. Estimate the percentage of students who were educated in Maryland and Virginia combined. A. 45.0% B. 4.5% C. 30.0% D. 11.4% E. 25.0%

24. A McDonald’s customer ordered a burger for 79¢, fries for $1.19, and a coke for $1.29. If the locality has a 5% sales tax, approximately what is the total bill? A. $3.00 B. $5.25 C. $4.00 D. $3.45

25. 40 30 20 10 Percent Change of Value In Stock ’99 ‘00 ‘01 | -10 -20 1990 ’91 ’92 ’93 94 ’ 95 96 ’ 97 ‘ 98 From the information provided about company XYZ, what appears to have occurred? A. The value of the stock reached its highest value in 1995. B. The value of the stock was still increasing in 1995, but began to decline the next year. C. The value of the stock increased in 1995 and the value also increased but at a slower rate in the next year. D. There is not enough information provided to answer the question.

26. 6) While on vacation in Las Vegas Jim wants to send a postcard to his grandmother in Fayetteville, AR. He remembers that her zip code begins with either “72” or “73”. How many 5-digit zip code possibilities begin with either “72” or “73”? A. 2000 B. 1998 C. 1000 D. 999 E. 10,000 7) If your score on a test is decreased by 30% and then increased by 35%, is the final result more or less than the original score? Explain your answer.

27. If you flip a coin five times, which of the following results is (are) more likely? Least likely? • The possible results of the flips, H or T, are in order from first flip to fifth flip. • Explain your answer. • HHHHT, TTTTT, HHHTT, THHTT, HHHHH • If you flip a coin five times, which of the following results is (are) most likely? Least likely? Explain your answer. • all H • all T • 3T and 2H • 4T and 1H • 4H and 1T

28. Sample tasks: Can both of these views be correct? Explain. In each graph there is a “bar” over $20,000 to $30,000. Do these two bars represent the same quantity? Explain.

30. Changing Health Care Costs Why are these graphs titled as “The Rise in Spending?” Assume that US citizens spent $10 billion on prescription drugs in 1989. a) Use the information in the graph on page 25 of the NYT article (8/11/02) to produce a graph of prescription drugs spending for 1989-2000. b) On the same axes produce a graph of the same $10 billion over the same period when subjected to inflation. c) Write 2-3 sentences interpreting the meaning of the two graphs.

31. Refer to the December 6 letter to the editor, Math skills aren’t great. • Find the increase in percent proficient. • Find the percent increase in the percent proficient. • Is the letter writer correct that the original article was wrong? Why • Is the letter writer correct or incorrect when he states, “going from 1 percent proficient tyo 3 percent proficient is an increase of 200 percent?” Why?

32. Refer to the December 6 letter to the editor, Math skills aren’t great. • Find the increase in percent proficient. • 1% X=3% x=300% • b) Find the percent increase in the percent proficient. • .01 x 3.00 or .01 x 300% = .03 • c) Is the letter writer correct that the original article was wrong? Why? • No, he did not correctly calculate the percent change. • d) Is the letter writer correct or incorrect when he states, “going from 1 percent proficient to 3 percent proficient is an increase of 200 percent?” Why? • No, it is an increase of 300%, not 200%

33. Find the increase in percent proficient. • 1% 3% • The percent proficient increased by two percentage points. • b) Find the percent increase in the percent proficient. • c) Is the letter writer correct that the original article was wrong? Why? • The letter writer was correct, but he needs to calm down a bit. It was a small, common mistake, but a mistake nonetheless. • d) Is the letter writer correct or incorrect when he states, “going from 1 percent proficient to 3 percent proficient is an increase of 200 percent?” Why? • He is correct. The editorial assumed that if the # tripled, it would mean it increased by 300%. What the editorial forgot to do was add on to the original # to the problem. • 1% 2% • X2 +1% 3% It is the same reason why a number that doubles increases only 100%

34. Find the increase in percent proficient. • Increase in percent: 3% - 1% = 2% increase • b) Find the percent increase in the percent proficient. • Percent increase: • c) Is the letter writer correct that the original article was wrong? Why? • Yes, because if the percent increase was to be 300% like the original article stated, the ending proficiency would need to be 4% instead of 3%. • Ex: • d) Is the letter writer correct or incorrect when he states, “going from 1 percent proficient to 3 percent proficient is an increase of 200 percent?” Why? • The letter writer is incorrect in making that statement due to a misuse of wording. The letter writer made an error in saying “increase of 200%,” when he should have said “it’s a percent increase of 200%.”

35. This letter to the editor appeared in the Arkansas Democrat-Gazette onApril 9, 2002. My children asked me how many ancestors and how many acts of these ancestors they are responsible for after reading and listening to the Razorbacks’ coaching dilemma. They have been taught that they are responsible for their own actions and sometimes the actions of their friends or even their parents. They just want to know how far this goes back. My daughter had visited the slave ship exhibit at one of our downtown museums and recognized a family name as being a builder of slave ships back in the 1500s in Britain. She also knew that another relative brought six slaves over to Jamestown in the 1600s. How much was she going to have to pay in retribution? Was she the only one responsible or were there others?

36. Before this got even more out of hand, we decided to do the math. Assuming four generations per century and only one child per family, that would be 19 generations. Two to the power of 19 would be 524,288 people who shared the responsibility. Then we started laughing at the total absurdity of the idea of one person today paying for the sins of another when there had been 524,288 people in between. And that wasn’t even counting brothers and sisters. Conclusion: Get a life. Forgive and forget all 524,288 of them. Analysis of the argument 1. How might the writer arrive at the conclusion of 524,288? 2. Are there other possible conclusions for the number of people in between the ancestor and the daughter?

37. The very forces that now press colleges to address issues of quantitative literacy were created by colleges and universities in the first place. Changes in society that demand widespread quantitative literacy arose in large measure from innovations of college graduates seeking greater effectiveness and efficiencies in the fabric of life and work. We must turn our creativity toward coping with our creations.