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F IVE A LIVE ! A C RITICAL T HINKING P ILOT P ROGRAM FOR M ATHEMATICS Jim Rutledge and Carol Weideman Faculty Co-Champions, Mathematics

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I NTRODUCTION QEP Critical Thinking Initiative at SPC 2011—Mathematics Math QEP Committee met during Spring and Summer, 2011 Decided to focus on problem solving as an over- arching theme and more specifically on critical thinking skills pertaining to problem solving.

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E XAMPLES OF CRITICAL THINKING SKILLS PERTINENT TO PROBLEM SOLVING Analyze data and search for patterns Analyze possible outcomes Construct a diagram Eliminate the impossible Guess, check and revise Identify relevant (and irrelevant) information Interpret graphical data Use lists or tables to order and display data

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V ISION, G OALS & SUPPORT Vision: Each math class (college-wide) would identify, illustrate and promote critical thinking skills as an integrated part of the curriculum. Goals: To impress upon students the importance of critical thinking and to give them enjoyable opportunities to exercise and develop these skills. Support: By intentionally addressing critical thinking in each class, faculty will underscore its importance and validate the college-wide effort.

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P ROGRAM CONCEPT & DEVELOPMENT Concept: Use the first five to ten minutes of certain classes to introduce a teaching/learning episode that illustrates a specific critical thinking skill and then provide students with the opportunity to exercise that skill in a problem- solving mode. Five Alive! pilot program implemented in Summer session and continued in Fall, 2011. MGF 1106—Carol Weideman (online, blended) MGF 1107—Jim Rutledge (on campus)

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F IVE A LIVE ! O VERVIEW —MGF 1107 MGF 1107 program consisted of three two-week segments (involving the first ten minutes of four class meetings), each of which focused on a specific critical thinking skill. Students worked in groups of two to solve the Five Alive! critical thinking challenges. Students had an opportunity to present their solutions to the class. Certificates of achievement were awarded at the end of the semester (based on achievement level).

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R ESULTS —S UMMER, 2011 Pretest average score: 1.6 (out of 10) Post-test average score: 4.0 (out of 10) Optional (but highly encouraged) participation No credit toward grade Three Five Alive! assignments—total of 15 achievement points possible: 4 students earned 10-15 points 19 students earned 5-9 points 1 student earned 1-4 points 5 students did not participate

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C HANGES FOR FALL, 2011 Survey at end of Summer indicated that some students objected to the fact that the Five Alive! assignments did not contribute to their grade. The MGF 1107 Fall pilot will award extra credit for Five Alive! achievement (15 points maximum; 800 normal points in semester). More emphasis will be put on having students record their critical thinking processes. This was done minimally by students in the Summer. More encouragement will be given to students to share their successful problem-solving efforts with the class as a whole.

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Five Alive! Critical Thinking SkillsPascal’s Triangle Critical thinking skill: Analyze data and search for patterns Application illustration The first several rows of Pascal’s Triangle are presented to Burt and Izzy and they are asked to determine the entries in the next row. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1

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Izzy says to Burt, “Well, Burt, it seems pretty obvious that the first and last entries in the next row will be 1’s since that is the case for each row.” “Good observation, Izzy,” replies Burt, “and it is also fairly obvious that the second and next-to-last entries will be 6’s since those entries always increase by one in each successive row. But what about the middle part of the row?” “Hmm.m.m…..” says Izzy. That’s a bit more difficult. How do you think they got the 10’s in the middle of the current bottom row?” “Aha!” exclaims Burt. “I see it! If you add the 4 and the 6 in the row above, you get 10 as the result in the row below!” And that pattern works for the other middle entries as well. So 1+4=5, and 4+6=10; in the same way, 6+4=10 and 4+1=5. And that’s how you get the middle entry values! That’s pretty exciting!” “Dude!” declares Izzy. “So the next row will have middle entry values of 1+5=6, 5+10=15, 10+10=20, 10+5=15, and 5+1=6.” “Awesome,” says Burt. “So the next row consists of the entries: 1 6 15 20 15 6 1”

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Five Alive! assignment Using an analytical approach similar to Burt and Izzy’s, determine the pattern involved in this variation of Pascal’s Triangle and determine the entries in the next row: 1 1 1 1 3 1 1 5 5 1 1 7 13 7 1 1 9 25 25 9 1 Entries in next row: _____________________________________________________________ In a paragraph, please describe the critical thinking process that led you to your solution. Specifically, please describe the conjectures that you made (including those that turned out to be incorrect) as you searched for the correct solution.

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F IVE A LIVE ! O VERVIEW —MGF 1106 MGF 1106 program was introduced in an online summer section. The program consisted of six discussion topics focused on course material and critical thinking; students worked in teams of 5-6 students. Each week a new topic was introduced. Each topic included several questions for student response. Certificates of achievement were awarded at the end of the semester (based on achievement level).

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MGF1106: T OPICS Course covers topics ranging from problem solving and critical thinking, logic, statistics, geometry Course begins with a focus on problem solving Five Alive Activities introduced with the concept of problem solving

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R ESULTS —S UMMER, 2011 o Six Five Alive! Activities incorporated as discussion topics in online course Each activity was worth 12 points, with lowest score dropped: 60 points out of 380 points 14 students earned 51-60 points 1 student earned 41-50 points 4 students earned 31-40 points 5 students earned less than 30 points (3 of these students failed the course) Anonymous student survey: 93% felt these activities reinforced the course concepts

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MGF1106: C HANGES FOR FALL, 2011 Course offered in blended format Pretest included as part of course orientation (mean = 53%) Posttest will be given at end of course MGF 1106 Fall pilot will award extra credit for Five Alive! achievement More emphasis will be put on having students record their critical thinking processes. More encouragement will be given to students to share their successful problem-solving efforts with the class as a whole.

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P OLYA ' S METHOD * 1. Understand the problem 2. Devise a plan to solve the problem 3. Carry out the plan 4. Check the results * From Polya’s book “How to Solve It,” published in 1945

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F IVE A LIVE ! P ROBLEM S OLVING A PPROACH 1. State the question clearly. Work with one problem at a time 2. Understand and translate. 3. Work out a plan (or plans) for solving. Identify assumptions and determine if they are reasonable. 4. Use the information provided to carry out the plan; make sure you have sufficient information. 5. Check the results. If there are alternative plans for solving check by using an alternative approach. If the results are not reasonable, you can go back to step 2 and try again. 6. State the results; infer only what the evidence implies. Discuss implications and consequences.

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F IVE A LIVE ! A CTIVITY : S TACK OF C UBES Identical blocks are stacked in the corner of the room as shown: How many of the blocks are not visible?

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S OLVING THE P ROBLEM Step 1: State the question clearly. Work with one problem at a time We want to determine the number of blocks we cannot see in the stack of cubes Step 2 : Understand and translate. Since it appears to be easy to calculate the blocks we can see, we could calculate the total number of blocks and subtract the visible blocks to find the hidden blocks. Alternatively we could calculate the total in each row and subtract the front visible row; repeat for each of the five row.

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S OLVING THE P ROBLEM Step 3: Work out a plan (or plans) for solving. Identify assumptions and determine if they are reasonable. Assumption; We assume that each row extends fully in the corner. Let x = total blocks in stack Let y = visible blocks Then: # of Hidden cubes = x – y

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S OLVING THE P ROBLEM Step 4: Use the information provided to carry out the plan. We’ll use the first approach. We’ll find x (total blocks) Top row has one block. Second row has 1 + 2 = 3 blocks Third row has 1 + 2 + 3 = 6 blocks Fourth row has 1 + 2 + 3 + 4 = 10 blocks Fifth row has 1 + 2 + 3 + 4 + 5 = 15 blocks Total blocks in the stack = 1 + 3 + 6 + 10 + 15 = 35 blocks So x = 35 blocks

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S OLVING THE P ROBLEM Step 4 (continued): Top row has one block. Second row has 1 + 2 = 3 blocks Third row has 1 + 2 + 3 = 6 blocks Fourth row has 1 + 2 + 3 + 4 = 10 blocks Fifth row has 1 + 2 + 3 + 4 + 5 = 15 blocks Notice another relationship among the rows: Blocks in row (n + 1) = (# in row n) + (n +1) Row 4 = # in Row 3 + 4 = 6 + 4 = 10 Row 5 = # in Row 4 + 5 = 10 + 5

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S OLVING THE P ROBLEM Step 4 (continued): Now we’ll find y (visible blocks) Top row has one visible block. Second row has 2 visible blocks Third row has 3 visible blocks Fourth row has 4 visible blocks Fifth row has 5 visible blocks Total visible blocks = 1 + 2 + 3 + 4 + 5 = 15 blocks So y = 15 blocks Total hidden blocks = x – y = 35 – 15 = 20 blocks

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S OLVING THE P ROBLEM Step 5: Check the results We can use the alternative plan to check our answer. How many hidden blocks in each row? Row 1 – none Row 2: 3 – 2 = 1 hidden Row 3: 6 – 3 = 3 hidden Row 4: 10 – 4 = 6 hidden Row 5: 15 – 5 = 10 Total Hidden : 1 + 3 + 6 + 10 = 20 blocks

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S OLVING THE P ROBLEM Step 6: State the results. Cubes stacked in a corner with 5 rows have 20 hidden blocks.

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T RY THIS YOURSELF ! If the stack had 6 rows, how many blocks are hidden? We know that stack with the five rows had 20 hidden so we only need to add the hidden blocks in the 6 th row. Solution: 20 + 15= 35 hidden blocks Notice that the hidden blocks a row are equal to the total blocks in the row above

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MGF1106 P ROBLEM S OLVING C HECKLIST

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R EUSABLE L EARNING O BJECT Stack of Cubes: SoftChalk Activity http://softchalkconnect.com/lesson/serve/LQFNbkp gxodjtY/html

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F IVE A LIVE ! G EOMETRY A CTIVITY Samantha is thinking of buying a circular hot tub 12ft in diameter, 4 ft deep and weighing 475 lbs. She wants to place the hot tub in a deck built to support 30,000 lb. Use π = 3.14. NOTE: Round each answer to the nearest whole unit. Can the deck support the hot tub?

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G EOMETRY F IVE A LIVE A CTIVITY Samantha is thinking of buying a circular hot tub 12ft in diameter, 4 ft deep and weighing 475 lbs. She wants to place the hot tub in a deck built to support 30,000 lb. 1.Determine the volume of water in the hot tub in cubic feet. 2.Determine the number of gallons of water the hot tub will hold. NOTE: 1 ft 3 = 7.5 gal. Hint: Volume = π r 2 hV = (3.14) (6) 2 (4) = 144 (3.14) = 452.16 ft 3 Hint: Radius = diameter/2 Gallons = 452.16 ft 3 * 7.5 gal = 3391.2 gallons

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G EOMETRY F IVE A LIVE A CTIVITY Samantha is thinking of buying a circular hot tub 12ft in diameter, 4 ft deep and weighing 475 lbs. She wants to place the hot tub in a deck built to support 30,000 lb. Use π = 3.14. NOTE: Round each answer to the nearest whole unit. 3.Determine the weight of the water in the hot tub. NOTE: Fresh water weighs about 8.35 lbs/gal. 4.Will the deck support the weight of the hot tub and the water? Support your answer. 5.Will the deck support the weight of the hot tub, water and four people, whose average weight is 115lb? Support your answer. Weight = 3392 gal (8.35 lbs/gal) = 28,316.52 Yes: Hot tub + Water = 475 + 28317 = 28,792 < 30,000 Yes: Hot tub + Water + 4(115) = 28,792 + 600 = 29,392 < 30,000

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F IVE A LIVE ! F UTURE P LANS Recruit other math faculty to incorporate activities Expand to other courses: STA2023, MAC1105 Use Five Alive! Activities as Discussion Topics in online, blended courses We welcome your feedback and input! Any Questions?

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L OGIC F IVE A LIVE A CTIVITY Conservative commentator Rush Limbaugh directed this passage at liberals and they way they think about crime. “Of course, liberals will argue that these actions (contemporary youth crime) can be laid at the foot of socioeconomic inequalities, or poverty. However, the Great Depression caused a level of poverty unknown to exist in America today, and yet I have been unable to find any accounts of crime waves sweeping our large cities. Let the liberals chew on that” (from See, I told You So, p. 83) We can write this passage as an argument:

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L OGIC F IVE A LIVE A CTIVITY Q UESTIONS es, Is the argument valid? Identify the standard form of the argument. The argument is valid: Law of Contraposition Where p = Poverty causes crime q = Crime waves swept American cities during the Great Depression

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